Uncertainty of a Physically Motivated Internal State Variable Plasticity and Damage Model (MSU DMG 1.0)

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Author(s): Kiran N. Solanki

Contents

Abstract

This report describes a methodology for efficient calibration of an internal state variable (ISV) material model that includes uncertainty. The model was developed to represent uncertainty in the material performance due to variations and inaccuracies in the model parameters. The model parameters comprise of constants collected from the literature (melting temperature, bulk modulus, shear modulus, etc.), constants derived from microscope images of microstructural features (voids, cracks, inclusion particles) in the material sample, and constants calibrated from experimental stress-strain data. The uncertainty in the model arises from inaccuracies in literature data, from variations in the measurements of the microstructures, and from inaccuracies in the stress-strain calibration data. The model calibration process is very computationally-intensive since variability distributions need to be calculated for the parameters in the model. The process involves Monte Carlo simulation with a large number of samples (10^5). For each sample, a function minimization problem is solved using a derivative-free method, requiring up to twenty seconds of solution time. As an estimate of the magnitude of the calculations, a ten-second solution time for each sample translates into more than eleven days of total process runtime on a serial machine. Thus, the process must be executed on a high performance parallel environment in order to obtain results within a reasonable period. The codes used in the process include Fortran routines for a material point simulator and an ABAQUS UMAT, MATLAB scripts for optimization and MatlabMPI for parallel execution on a Linux cluster. The computational methodology we describe enables the model calibration with uncertainty to complete over a weekend on 16 processors.

Introduction

Understanding the effect of material microstructural heterogeneities and the associated mechanical property uncertainties in the design phase is pivotal not only in terms of successful development of reliable, safe, and economical systems but also for the development of a new generation of lightweight designs [1]. Engineering systems contain different kinds of uncertainties found in material and component structures, computational models, input variables, and constraints [2]. Potential sources of uncertainty in a system include human errors, manufacturing or processing variations, operating condition variations, inaccurate or insufficient data, assumptions and idealizations, and lack of knowledge [3]. Since engineering materials are complex, hierarchical, heterogeneous systems, adopting a deterministic approach to materials design may be limiting. First, microstructure is inherently random at different scales [4]. Second, parameters of a given model are subject to variations associated with variations of material microstructure from specimen to specimen [5]. Furthermore, uncertainty should be associated with model-based predictions for several reasons. Models inevitably incorporate assumptions and approximations that impact the precision and accuracy of predictions [6]. Uncertainty may increase when a model is used near the limits of its intended domain of applicability and when information propagates through a series of models. Also, to facilitate exploration of a broad design space, approximate or surrogate models may be utilized, but fidelity may be sacrificed for computational efficiency. Experimental data for conditioning or validating approximate (or detailed) models may be sparse and may be affected by measurement errors. Also, uncertainty can be associated with the structural member tolerance differences and morphologies of realized material microstructure due to variations in processing history. Often, it is expensive or impossible to remove and measure these sources of variability, but their impact on model predictions and final system performance can be profound. As suggested by Horstemeyer et al. (2005)[5], a small variability (~1%) in the microstructure can result in a very large (~13%) variation in the failure stress.

Figure 1 Procedure for studying the effect of material microstructure variability and loading conditions on material response and their failure mechanisms.

Most of the structural materials used for component design exhibit material property variations which could be linked to microstructural details. These microstructural variations are a resultant of material processing such as rolling or extrusion undergoing extensive plastic deformation. During material processing, cavities and second phase particles are elongated (aspect ratio distributions) and oriented (distribution), in such a way that the larger axis tends to align itself along the maximum principal load direction. This inclusion distribution has been shown to subsequently generate variations in structural responses and strongly influence the ductile fracture of structural steels, where macroscopic compliances evolve differently in orthogonal directions [7][8][9]. This orientation distribution dependent ductility leads to variability in the damage evolution in quasistatic fracture response [10] and ballistic impact resistance. The void nucleation and void growth evolutions can also depend upon the initial grain boundary misorientation distribution and grain morphology[11].

For designing reliable and robust engineering systems, designers seek materials with optimal values of properties such as strength and toughness. To achieve such a material, appropriate models should relate deformation to the key microstructures such as particle size, interfacial strength, and spacing[12]. In addition, models can account for the uncertainty of material properties and microstructural characteristics[5]. With the help of such microstructure-property relationship constitutive models, it is possible to relate the mechanical properties of interest, such as stress, strain, and toughness, to key microstructures such as particle size and spacing, interfacial strength, and grain size.

In this study, internal state variable (ISV) based theory is utilized to capture the microstructure-property relationships. ISV theory is often employed to solve engineering problems by linking lower length scale information[3][13][14][12][15][16][17][18], which are formulated at the macroscale level. The ISVs reflect the lower spatial size scale microstructural rearrangements so that history effects can be modeled. Onat et al. (1988) demonstrated that tensorial ISVs constitute a natural tool for the representation of internal structure and its orientation. ISVs also provide a convenient measure of the degree of anisotropy present in the material. Studies have shown that representations based on the notion of state and on the differential equations that govern the evolution of state have definite advantages over other methods of representation (i.e. integral representation). Some examples of the application of ISVs in metals are hysteresis due to plastic deformation or phase transition and fatigue and fracture [5]. McDowell also noted that some of these processes occur so slowly and so near equilibrium that common models forego description of nonequilibrium aspects of dissipation (e.g. grain growth). The ISVs constitutive theory offers an in-depth basis for incorporating irreversible, path dependent behavior that can be informed by experiments, computational materials science and micromechanics [5]. One pure advantage of an ISV model is the ability to alleviate boundary conditions when instituting the model in a finite element (FE) environment. This asset is due to the model’s ability to predict path history dependence and complex boundary value problems.

In a recent investigation, one of the authors’ applied a structure-property analysis based on the internal state variable formulation to an automotive component made of A356 cast aluminum alloy in which a hierarchical multiscale modeling methodology was developed [19][14][20][12][15][16][21][22]. Through microstructure-property relationship modeling, they were able to correctly predict the material failure mode of the component under multiple static load cases; in contrast, failure predictions that were based solely on stress field or inclusion analysis using optical imaging did not agree with the results of physical experiments. In the subsequent redesign guided by microstructure-property relationship material modeling, the component’s fatigue performance was improved by more than two fold while its weight was reduced by as much as 25%. For a summary of recent progress in microstructure-property relationship material modeling and simulations, the reader is referred to Horstemeyer (2001)[16]; Horstemeyer et al. (2003b)[22]; Graham-Brady et al. (2006)[23].

The rational and motivation of this study is to understand the effect of material internal heterogeneity and boundary conditions on localized damage and its progression mechanism. The failure mechanism studied in this chapter is a result of void-nucleation, growth and coalescences, which are modeled using an physically based internal state variable form of microstructure-property model, which was initially developed for plasticity by Bammann et al. (1993)[18] and later modified to incorporate damage due to void-nucleation, growth and coalescences by Horstemeyer et al. (1999, 2000)[14][12][15]. The overall procedure to study the effect of material microstructural variability and boundary conditions on the material response and failure mechanism can be outlined as depicted in Figure 1. First, the experimental uncertainties due to both systematic and bias are calculated along with uncertainties in the microstructure features in the model. Second, the microstructure features with their variability, the experimental stress-strain curves with their variability and the model calibration routine are used to calibrate material parameters along with their variability. The calibration routine used here is a material point simulator. Finally, a user material subroutine is developed, implemented, and used to predict material mechanical responses and material failure mechanisms along with their respective uncertainties. One of the most important contributions the authors believe is to show how internal microstructural clustering affects strain to failure. This report is structured as follows. Section 2 describes the microstructure-property relationship modeling framework used in the study. The model contains internal state variables to account for localized damage and damage progression. The model is applied to wrought 7075-T651 aluminum alloy. Section 3 describes the mechanical and microscope imaging experiments that were undertaken to characterize this material. Stress-strain responses of the material under various loading conditions were calculated from the experimental measurements. Microstructural features such as average grain size, mean values of the particle area, particle area fraction, aspect ratio and nearest neighbor distances were also established. The associated uncertainties were also calculated. Section 4 describes the calibration of the ISV model with consideration for uncertainty. Here model calibration means, fitting the model parameters using a material point simulator and calculating the uncertainty in the model predictions using a Monte-Carlo technique with a sample size of 106. This process is computationally-intensive since the variability distribution for each model parameter is calculated. This process was executed on a Linux cluster to reduce the runtime. Section 5 presents sample results obtained using this procedure and the interpretation of these results. This section also describes the verification of the model through simulation using a finite element software and the validation of the model through additional experiments. Section 6 gives the concluding remarks.

Microstructure-Property Relationships

Performance and failure of metallic components under static or dynamic loads are strongly influenced by the microstructural features (e.g., voids, cracks, and inclusion particles) of the material. These features depend on the type of material, the process by which it is transformed into a product, and the environment in which the product is used. Traditional design techniques that rely on factors of safety and stress failure criteria (e.g., von-Mises, Tresca) do not always result in a safe design. This inadequate design can arise, because material microstructures and the loading history can affect the response characteristics of the component and cause its premature failure.

The microstructure-property relationship modeling framework used for our work is based on that of Bammann et al. (1984, 1987, 1989, 1990, 1993, 1995, 1996)[24][25][26][17][18][27]. The constitutive model used here contains physically motivated internal state variables to account for the motion of dislocations and the evolution of dislocation structures. The framework also accounts for stress-state-dependent damage evolution. The pertinent equations in this model are denoted by the rate of change of the observable and internal state variables. In this chapter, however we only briefly discuss their relation to the uncertainty analysis. For more details about the equations, the reader can refer to Horstemeyer et al. (1999, 2000, 2001)[14][12][15][16]. The equations used within the context of the finite element method are the rate of change of the observable and internal state variables given by


\overset{\circ}{\underset{-}{\sigma}} = {\lambda}\left(1-{\Phi}\right )tr\left({\underset{-}{D}}^e\right) \underset{-}{I} + 2{\mu}\left( 1-{\Phi}\right )){\underset{-}{D}}^e - \left (\frac{\dot{\phi}}{1-{\Phi}} \right )\overset{\ }{\underset{-}{\sigma}}                     (1)

where \overset{\ }{\underset{-}{\sigma}} and \overset{\circ}{\underset{-}{\sigma}} are the Cauchy stress and the co-rotational rate of the Cauchy stress, respectively; \Phi is an ISV that represents the damage state with \dot{\phi} representing its material time derivative; \lambda and \mu are the elastic Lame constants; {\underset{-}{D}}^e is the elastic deformation tensor; and is the second-order identity tensor. The underscore symbol indicates a second rank tensor. The plastic deformation tensor or inelastic flow rule,{\underset{-}{D}}^p , is given by the relationship


{\underset{-}{D}}^p = f(T)\sin h\quad \left \lbrace \frac{\|{\underset{-}{\sigma}}^'-\underset{-}{\alpha} \|-\quad \left \lbrack R +Y \left ( T \right ) \right \rbrack \left( 1-{\Phi}\right ) }{V \left ( T \right )\left( 1-{\Phi}\right ) } \right \rbrace \frac{{\underset{-}{\sigma}}^'-\underset{-}{\alpha}}{\|{\underset{-}{\sigma}}^'-\underset{-}{\alpha}\|}              (2)


where {\underset{-}{\sigma}}^' is the deviatoric part of stress tensor; T is temperature in Kelvin;\underset{-}{\alpha} is the kinematic hardening (an ISV reflecting the effect of anisotropic dislocation density); and R is the isotropic hardening (an ISV reflecting the effect of global dislocation density). The function V(T) determines the magnitude of rate-dependence on yielding; f(T) determines when the rate-dependence affects initial yielding; and Y(T) is the rate-independent yield stress. Functions V(T), Y(T), and f(T) are related to yielding with Arrhenius-type temperature dependence and are given as


V \left ( T \right ) = C_1e^{\left ( -C_2/T\ \right)},          Y \left ( T \right ) = C_3e^{\left ( -C_4/T\ \right)},         f(T) = C_5e^{\left ( -C_6/T\ \right)}            (3)


where C_1 through C_6 are the yield stress related material parameters that are obtained from isothermal compression tests with variations in temperature and strain rate.

The co-rotational rate of the kinematic hardening, \overset{\circ}{\underset{-}{\alpha}} and the material time derivative of isotropic hardening,\dot{R} are expressed in a hardening-recovery format as


\overset{\circ}{\underset{-}{\alpha}} = \quad \left \lbrace h \left ( T \right ){\underset{-}{D}}^p-\quad \left \lbrack \sqrt{\frac{2}{3}}r_d \left (T \right ) \|{\underset{-}{D}}^p\|+ r_s \left (T \right) \right \rbrack \|\underset{-}{\alpha}\|\underset{-}{\alpha}\right \rbrace {\quad \left \lbrack \frac{{DCS}_0}{DCS} \right \rbrack }^Z            (4)


\dot{R} = \quad \left \lbrace H \left ( T \right ){\underset{-}{D}}^p-\quad \left \lbrack \sqrt{\frac{2}{3}}R_d \left (T \right ) \|{\underset{-}{D}}^p\|+ R_s \left (T \right) \right \rbrack R^2 \right \rbrace {\quad \left \lbrack \frac{{DCS}_0}{DCS} \right \rbrack }^Z            (5)


where DCS_0, DCS, and z parameters capture the microstructure effect of grain size. The dislocation populations and morphology within crystallographic materials exhibit two types of recovery. In Eqs. 4 and 5, r_d(T) and R_d(T) are scalar functions of temperature that describe dynamic recovery, r_s(T) and R_s(T) are scalar functions that describe thermal (static) recovery, whereas h(T) and H(T) represent anisotropic and isotropic hardening modulus, respectively. These functions are calculated as


r_d \left(T \right) = C_7 \left \lbrack 1 + C_a \left ( \frac{4}{27} - \frac{{J_3}^{'2}}{{J_2}^{'3}} \right )-C_b {\left ( \frac{{J_3}^'}{{J_2}^'} \right )}^{\frac{3}{2}} \right \rbrack e^{\left ( -C_8/T\ \right)}                  (6a)


R_d \left(T \right) = C_{13} \left \lbrack 1 + C_a \left ( \frac{4}{27} - \frac{{J_3}^{'2}}{{J_2}^{'3}} \right )-C_b {\left ( \frac{{J_3}^'}{{J_2}^'} \right )}^{\frac{3}{2}} \right \rbrack e^{\left ( -C_{14}/T\ \right)}                  (6b)


r_s \left(T \right) = C_{11}e^{\left ( -C_{12}/T\ \right)}                  (6c)


R_s \left(T \right) = C_{17}e^{\left ( -C_{18}/T\ \right)}                  (6d)


h \left(T \right) = C_9 \left \lbrack 1 + C_a \left ( \frac{4}{27} - \frac{{J_3}^{'2}}{{J_2}^{'3}} \right )-C_b {\left ( \frac{{J_3}^'}{{J_2}^'} \right )}^{\frac{3}{2}} \right \rbrack e^{\left ( -C_8/T\ \right)}-C_{10}T                  (6e)


H \left(T \right) = C_{15} \left \lbrack 1 + C_{22} \left ( \frac{4}{27} - \frac{{J_3}^{'2}}{{J_2}^{'3}} \right )-C_b {\left ( \frac{{J_3}^'}{{J_2}^'} \right )}^{\frac{3}{2}} \right \rbrack e^{\left ( -C_8/T\ \right)}-C_{16}T                  (6f)


where {J_2}^'=\frac{1}{2}{({\overset{\circ}{\underset{-}{\sigma}}-{\underset{-}{\alpha}}})}^2 ,{J_3}^'=\frac{1}{3}{({\overset{\circ}{\underset{-}{\sigma}}-{\underset{-}{\alpha}}})}^3 , C_7 through C_{12} are the material plasticity parameters related to kinematic hardening and recovery terms, C_{13} through C_{18}are the material plasticity parameters related to isotropic hardening and recovery terms, whereas Ca and Cb are the material plasticity parameters related to dynamic recovery and anisotropic hardening terms, respectively. Constants C_1 through C_{18} are determined from macroscale experiments at different temperatures and strain rates. The damage variable, represents the damage fraction of material within a continuum element. The mechanical properties of a material depend upon the amount and type of microdefects within its structure. Deformation changes these microdefects, and when the number of microdefects accumulates, damage is said to have grown. The three components of damage progression mechanism are void nucleation, growth and coalescence from second phase particles and pores. In this regard, the material time derivative of damage,\dot{\phi} is expressed as


\dot{\phi} =  \quad \left \lbrack \dot{{\phi}}_{particles} + \dot{{\phi}}_{pores} \right \rbrack C + \quad \left \lbrack {\phi}_{particles} + {\phi}_{pores} \right \rbrack \dot{C}                   (7)


where {\Phi}_{particles} represents void growth from particle debonding and fracture;{\Phi}_{pores} represents void growth from pores; with \dot{{\phi}}_{particles} and \dot{{\phi}}_{pores} representing their respective time derivatives; parameter c represents the void coalescence, or void interaction, that is indicative of pore-pore and particle-pore interactions with \dot{c} as its time derivative. The particle- and pore-based void growth rate and the void-coalescence rate equations are given as


\dot{\phi}_{particles} = \dot{\eta} {\nu} + {\eta}\dot{\nu}                  (8a)


\dot{\phi}_{pores} =  \quad \left \lbrack \frac{1}{{\left ( 1 - {\phi}_{pores} \right )}^m}-\left ( 1 - {\phi}_{pores} \right ) \right \rbrack\sinh\quad \left \lbrack \frac{2 \left ( 2^{V \left (T \right )}/{Y \left( T \right)}^{-1}\ \right ){{\sigma}_H}}{ \left ( 2^{V \left (T \right )}/{Y \left( T \right)}^{+1}\ \right ){{\sigma}_{vm}}} \right \rbrack \|{\underset{-}{D}}^p\|                  (8b)


\dot{c} =\quad \left \lbrack Cd_1 +Cd_2 \left ({\eta}\dot{\nu} + \dot{\eta} {\nu} \right ) \right \rbrack e^{\left ( C_{CT}T \right )} {\left ( DCS_0/DCS \ \right)}^Z                  (8c)


where {\nu} is the void growth; \eta is the void nucleation, whereas {\sigma}_H and {\sigma}_vm are the hydrostatic and von Mises stresses, respectively. The parameters Cd1 and Cd2 are related to first and second normalized nearest neighbor distance parameters, respectively, and C_CT is the void-coalescence temperature dependent parameter. The void nucleation rate and void growth rate are given as


\dot{\eta} = \|{\underset{-}{D}}^p\|\frac{C_{coeff}d^{\frac{1}{2}}}{K_{IC}f^{\frac{1}{3}}}{\eta} \quad \left \lbrack a \left ( \frac{4}{27} - \frac{{J_3}^2}{{J_2}^3} \right ) + b \frac{J_3}{{J_2}^{\frac{3}{2}}} + c \left \Vert \frac{I_1}{\sqrt{J_2}} \right \| \right \rbrack e^{ \left (-C_{{\eta}T}/T\ \right)}                  (9a)


\dot{\nu} =  \frac{\sqrt{3}R_0}{2\left (1-m \right)}\quad \left \lbrack \sinh \left ( \sqrt{3}\left (1-m \right)\frac{\sqrt{2}I_1}{3\sqrt{J_2}} \right ) \right \rbrack \|{\underset{-}{D}}^p\|                  (9b)


where C_coeff is a material constant that scales the response as a function of initial conditions; d is the particle size; K_IC is the fracture toughness; f is the volume fraction of second-phase particles; C_nT is the void nucleation temperature dependent parameter;I_1,J_2, and J_3 are the independent stress invariants; m void growth constant; R_0is the initial void radius, whereas material constants a, b, and c are the void-nucleation constants that are determined from different stress states (i.e., a is found from a torsion test, while b and c are determined from tension and compression tests, with all three having units of stress). The time integral form of Eq. 7 is used as the damage state. Based on this ISV model, material failure is assumed to occur when Eq. 7 reaches unity (\phi->1.0 ) within a finite element. For all practical purposes, material failure can be assumed at a much smaller value (safe limit) of \phi as the damage increases very rapidly to 1.0 shortly after \phi reaches a small percentage. The mechanical properties of a material depend upon the amount and type of microdefects within its structure. Deformation changes these microdefects, and when the number of microdefects accumulates, the damage state is said to have grown. By including damage, \phi as an ISV, different forms of damage rules can be incorporated easily into the constitutive framework. In summary, \alpha,R, \sigma , \phi ,c , v,and \eta in Eqs. 1 through 9 represent the ISVs in this microstructure-property relationship material model.

Material Mechanical Responses, Microstructure Characterizations, and their Uncertainties

The microstructure of a typical metallic material contains a large number of microdefects such as microcracks, dislocations, pores, and decohesions. Some of these defects are induced during the manufacturing process and are present before the material is subjected to mechanical loads and thermal fields. In general, these defects are small and distributed throughout most of the volume. In this report, we focus on wrought 7075-T651 aluminum alloy. The 7075-T651 aluminum is a wrought product with a relatively high yield strength and good ductility. As displayed by the triplanar optical micrographs of the plate concerned in the current investigation shown in Figure 2, the grains of this wrought alloy were found to be pancake shaped and aligned in the rolling direction of the wrought plate.

Figure 2. Triplanar optical micrograph illustrating the grain structure and orientation of 7075-T651 aluminum alloy.
Figure 3. Microstructure of unstrained 7075-T651 aluminum alloy (Harris, 2006).


An optical microscope image of the wrought alloy, as shown in Figure 3 displays typical 7075-T651 aluminum microstructure in the untested condition. The alloy contains two main types of primary particles: Iron rich particles ({Al}_6(Fe,Mn), {Al}_3Fe, \alphaAl(Fe,Mn,Si) and {Al}_7{Cu}_2Fe); and silicon compound particles ({Mg}_2Si). The iron-rich intermetallic particles are seen in the optical micrograph (Figure 3) as light grey particles and the Mg2Si intermetallics are shown as the dark particles. The particle size, nearest neighbor distance, and aspect ratio for the primary distributions for the iron-rich and ({Mg}_2Si) intermetallics in the untested condition were tabulated from a 5.75 {mm}^2 area of material for each of the orientations and are shown in Figure 4. Figure 4a displays the area size of the particles, Figure 4b displays the nearest neighbor distance of the particles and Figure 4c displays the aspect ratio distributions of the particles. The mean values of the particle area, area fraction, aspect ratio and nearest neighbor distances are displayed in Table 1. In addition to the intermetallic particle stereography, the average grain size was determined by EBSD analysis taken in each of the directions and the results listed in Table 1.

Table 1 Metallographic analysis of virgin aluminum 7075-T651 alloy: mean area particle fraction, particle size, nearest neighbor distance and grain size.
Figure 4(a)Stereological comparison of distributions of intermetallic particles (large) in 7075-T651 aluminum alloy for the longitudinal (L) direction: a) particle area.
Figure 4(b)Stereological comparison of distributions of intermetallic particles (large) in 7075-T651 aluminum alloy for the longitudinal (L) direction: b) nearest neighbor distance.
Figure 4(c)Stereological comparison of distributions of intermetallic particles (large) in 7075-T651 aluminum alloy for the longitudinal (L) direction: c)aspect ratio.

Figure 4 Stereological comparison of distributions of intermetallic particles (large) in 7075-T651 aluminum alloy for the longitudinal (L) direction: a) particle area, b) nearest neighbor distance, c) aspect ratio. In order to properly model the 7075-T651 alloy, standard monotonic experiments were performed (ASTM E8). Three different types of monotonic experiments (tensile, compression and torsion) were performed along the longitudinal direction. All experimental specimens were machined 2.54 mm from the rolling surface of the plate. The tension, compression, and torsion tests were performed with a strain rate of 0.001/sec in an ambient laboratory environment.

Table 2 Accuracy of instruments used to measure load-strain in monotonic tension, compression, and torsion tests.

Table 2 Accuracy of instruments used to measure load-strain in monotonic tension, compression, and torsion tests.

The experimental stress/strain curves were tabulated by taking load and strain values and by using the nominal cross-sectional area. In order to quantify random uncertainties of measured quantities, three specimens for each direction of the alloy were tested. Table 2 shows the accuracy related to different instruments used to measure load-strain curve. The experimental uncertainties are calculated based on systematic and random uncertainties in the measured quantities such as force, strain, and specimen sizes (Eqs.10-12) [28].


U_E =  \sqrt{{U_r}^2 + {U_s}^2}                   (10)


U_r =  2 \sqrt{\frac{1}{M-1}\sum_{i=1}^M {{\left ( r_i - r_{mean} \right)}^2}}                  (11)


where Ur is random uncertainty, and Us is systematic uncertainty The random uncertainty in experimentally measured quantities ri (force and strain) for M different tests is given by


The systematic uncertainty in experimentally measured quantities ri (force and strain) for M different tests is given by


U_s = r_i \sqrt{{U_L}^2 + {U_{daq}}^2}                   (12)


where UL is uncertainty in the load cell or extensometer or strain gauges, and Udaq is the uncertainty in the data acquisition. The tensile tests were conducted using a Tenuis Olson type tensile machine. The tests were conducted in constant cross-head control, with a speed of 5 inches per minute. An MTS knife blade axial extensometer with a 2 inch gage length was used for the strain measurement and was set at a full scale of 25% strain and calibrated to better than 0.25% through the full-scale range. The load cell was calibrated to within 0.25% error reading through the full-scale range. The stress strain data was collected on a System 5000 Data Acquisition system. Similarly, compression and torsional tests were also performed. Figure 7 shows mechanical responses of wrought 7075 aluminum alloy under tension, compression, and torsional loadings with their variability from mechanical experiments. From experimentally measured mechanical response, it was found that the variations in elongation to failure are about \pm5.2%, \pm6.3%, and \pm6.5% for torsion type loading, tensile loading, and compressive loading respectively.

Figure 7 Material mechanical response under (a) tension.
Figure 7 Material mechanical response under (c) torsion loadings with their variability.
Figure 7 Material mechanical response under (b) compression.


Figure 7 Material mechanical response under (a) tension, (b) compression, and (c) torsion loadings with their variability.

Additional tests were performed to further characterize the mechanical behavior (Bauschinger effects) of the wrought alloy. The Bauschinger effect is interpreted as anisotropic “yielding” that arises upon reverse loading from internal backstresses that are attributed to dislocations accumulating at obstacles. To quantify the Bauschinger effect, cylindrical low cycle fatigue type specimens with a uniform gage length based on ASTM standard E606 were used. Specimens with an outer diameter of 10.135 mm were used to test the 7075-T651 and were machined from the longitudinal direction of the two-inch-thick plate. The strain rate was 0.001/sec and the temperature was ambient. Two types of experiments to observe the Bauschinger effect were conducted. First the cylindrical specimens were prestrained in tension, and then uniaxially reloaded in compression. The second type included a different set of specimens that were prestrained in compression, and then uniaxially reloaded in tension. Figure 8 shows the Bauschinger behavior of the wrought 7075 aluminum alloy for tension followed by compression and compression followed by tension with their variability from mechanical experiments.

Figure 8 Material Bauschinger behavior under (a) tension followed by compression.
Figure 8 Material Bauschinger behavior under (b) compression followed by tension with their variability.


Figure 8 Material Bauschinger behavior under (a) tension followed by compression, and (b) compression followed by tension with their variability.Further experimental tests for the model calibration included interrupted tensile experiments in the longitudinal direction to determine the primary void nucleation rate and damage constants for the ISV model. As confirmed by Fabregue and Pardoen (2008), obtaining interrupted cracked or debonded sub-micron particles is not feasible and as such, the damage rate was assumed to be the same as the primary particles. Tensile specimens were loaded monotonically to pre-determined strain levels and sectioned for void and/or crack density quantification. The cracked and debonded primary particles were then quantified as a function of effective strain similar to Dighe et al. (2002)[29].

Model Calibration Strategy

Figure 9 Flowchart of process to fit the parameters of the ISV model. Initial model constants are refined by an optimization algorithm until the model curves correlate well with experimental stress-strain data.

In this section, we discuss a model calibration strategy for the structure-property ISV model (described in Section2) with its associated variability. The strategy used here differs from the classical uncertainty calibration technique. For the purposes of this study, model calibration with uncertainty consists of two steps: (a) fitting the model parameters using a material point simulator, and (b) calculating the associated uncertainties using a Monte-Carlo technique with a sample size of 10^5. A flowchart of the process to fit the model parameters is depicted by Figure 9, while a screenshot of the software implementation is displayed by Figure 10. We coded the material point simulator and the ISV model in Fortran, and we used MATLAB (The MathWorks Inc., 2008)[30] to implement the fitting routine and the graphical user interface.

Figure 10 Snapshot of the software for model fitting, showing good correlation of the experimental stress-strain data (discrete points) and the corresponding model curves.
Figure 11 Snapshot of the software showing the plots for one sample of the Monte Carlo study: the experimental data, the perturbed data, the model curve after perturbing the constants, and the model curve after the re-fitting the parameters.

The Monte Carlo technique to calculate parameter variability proceeds as follows. A sample is generated by adding random perturbations to the constants in the model and to the calibration data, where the perturbations are within the ranges of the variability of the constants and the data, respectively. The parameters in the model are then re-fitted for this sample. The procedure is repeated for 10^5 samples, and the distributions of the values for the parameters are calculated. Re-fitting the parameters may take up to twenty (20) seconds. Assuming a lower bound of 10 seconds for each sample, the aggregate execution time for 10^5 samples is almost 5.8 days on a single-processor machine. We utilized a toolkit created by one of the authors to parallelize the computation of the samples on a Linux cluster [31]. The toolkit incorporates dynamic load balancing to redistribute the samples among the participating processors in case of load imbalance due to differences in the execution times of re-fitting the model parameters. An example summary result for one sample is illustrated by Figure 11, while an example distribution of the values of a parameter is shown by Figure 12.

Figure 12a Snapshots of the software showing the distribution of a model parameter.
Figure 12b Snapshots of the software showing the distribution of a state variable.


Figure 12 Snapshots of the software showing the distribution of a model parameter and the distribution of a state variable .

Results, Validation, and Verification

In this section, we present results of calibrating the microstructure-property ISV model (Eqs. 1 to 9), along with its associated variabilities. The calibration strategy is described in the previous section. We calibrated the model with experimentally measured mechanical responses from tension, compression, torsion, tension followed by compression, and compression followed by tension and their variabilities. Along with mechanical responses, we also used material microstructure characteristics (particle size, particle orientation, void size, void orientation, grain size, and nearest neighbor distance) and their variability. To verify the calibrated model, it was implemented in a user material model routine in commercially available finite element code ABAQUS. The results of the verification simulations were compared with the experimental data. Finally, a series of validation experiments were performed to test the model’s prediction capability.

4.1 Model Calibration The microstructure-property based ISV model, Eqs. 1 to 6, includes plasticity material parameters related to the yield stress, the kinematic and the isotropic hardenings, and the shear and the bulk moduli along with their temperature and strain-rate dependencies, all of which were calibrated using experimentally measured mechanical responses and microstructural features, as shown in Table 3. Equations 7-9 include the damage progression, which is multiplicatively decomposed into void-nucleation, void-growth, and void-coalescence variables. The microstructure characteristic parameters are calibrated in conjunction with plasticity parameters using mechanical response and microstructure features as shown in Table 4. First, uncertainty in calibration data were calculated, then the model was fitted for a given set of mechanical responses and microstructural detail. The sampling distributions for microstructural features are shown in Figure 4. For mechanical responses we used a normal distribution with three standard deviations derived from the uncertainty calculations using the propagation rule [28].

Table 3 Microstructure-property (elastic-plastic) model parameters for 7075-T651 aluminum alloys

Table 3 Microstructure-property (elastic-plastic) model parameters for 7075-T651 aluminum alloys.

Table 4 Microstructure-property (damage) model parameters for 7075-T651aluminum alloys

Table 4 Microstructure-property (damage) model parameters for 7075-T651aluminum alloys.


The uncertainty distributions calculated during the calibration process using the Monte Carlo method related to the material and microstructural parameters of microstructure-property model are shown in Figures 9 to 15. Several observations can be drawn from these results. One can see that in almost every case the material mechanical response and microstructural parameters have the same distribution shape (Gaussian) but different uncertainty levels (the spread of distribution). This could be due to the different sensitivity levels of each model parameter in the constitutive relationship for this material.

Figure 9a Uncertainty distribution of material parameters related to the yield stress yield stress parameter which determines the rate independent yield stress.
Figure 9b Uncertainty distribution of material parameters related to the yield stress yield stress parameter which determines the transition strain rate from rate independent to dependent.


Figure 9 Uncertainty distribution of material parameters related to the yield stress (a) yield stress parameter which determines the rate independent yield stress, and (b) yield stress parameter which determines the transition strain rate from rate independent to dependent.

Figure 10a Uncertainty distribution of material parameters related to the isotropic hardening isotropic hardening parameter which describes the isotropic dynamic recovery.
Figure 10b Uncertainty distribution of material parameters related to the isotropic hardening isotropic hardening parameter which describes the isotropic hardening modulus.


Figure 10 Uncertainty distribution of material parameters related to the isotropic hardening (a) isotropic hardening parameter which describes the isotropic dynamic recovery, and (b) isotropic hardening parameter which describes the isotropic hardening modulus.

Figure 11a Uncertainty distribution of material parameters related to the kinematic hardening kinematic hardening parameter which describes the kinematic dynamic recovery.
Figure 11b Uncertainty distribution of material parameters related to the kinematic hardening kinematic hardening parameter which describes the kinematic hardening modulus.


Figure 11 Uncertainty distribution of material parameters related to the kinematic hardening (a) kinematic hardening parameter which describes the kinematic dynamic recovery, and (b) kinematic hardening parameter which describes the kinematic hardening modulus.

Figure 12a Uncertainty distribution of material parameters related to the void nucleation void-nucleation exponent coefficient
Figure 12b Uncertainty distribution of material parameters related to the void nucleation void-nucleation parameter related nucleation density under torsion.
Figure 12c Uncertainty distribution of material parameters related to void-nucleation parameters related to tension.
Figure 12d Uncertainty distribution of material parameters related to void-nucleation parameters related to compression void nucleation densities.


Figure 12 Uncertainty distribution of material parameters related to the void nucleation (a) void-nucleation exponent coefficient, (b) void-nucleation parameter related nucleation density under torsion, and (c-d) are void-nucleation parameters related to tension and compression void nucleation densities.

Figure 13a Uncertainty distribution of material parameters related to the void coalescence void coalescence parameter related to void sheeting.
Figure 13b Uncertainty distribution of material parameters related to the void coalescence void coalescence parameter related to void impingement.


Figure 13 Uncertainty distribution of material parameters related to the void coalescence (a) void coalescence parameter related to void sheeting, and (b) void coalescence parameter related to void impingement.

Figure 14a Uncertainty distribution of material parameters related to the temperature dependency on the void nucleation.
Figure 14a Uncertainty distribution of material parameters related to the temperature dependency on the void coalescence.


Figure 14 Uncertainty distribution of material parameters related to (a) the temperature dependency on the void nucleation, and (b) the temperature dependency on the void coalescence.

Figure 15 Uncertainty distribution of material parameters related to the material texture.


Figure 15 Uncertainty distribution of material parameters related to the material texture.

Having calibrated the model parameters and quantified the uncertainty associated with theses parameters; comparisons were made of stress evolution as a function of strain for different stress states to predict strain to failure and the work hardening behavior. Figures 16-17 show the effective stress–strain response comparisons for different stress states and also associate variability. As Figures 16-17 show the model is able to compare/predict the elongation to failure with a given level of uncertainty in initial microstructural clustering. It was found that the variations in elongation to failure are about ±7.0%, ±8.1%, and ±9.75% for torsion loading, tensile loading, and compressive loading respectively due to initial material microstructural heterogeneities.

Figure 16a Comparison of experimental tension, compression and torsion data with a microstructure-property model. The experiments were performed at a quasi-static strain rate Material mechanical response under tension.
Figure 16c Comparison of experimental tension, compression and torsion data with a microstructure-property model. The experiments were performed at a quasi-static strain rate Material mechanical response under compression.
Figure 16b Comparison of experimental tension, compression and torsion data with a microstructure-property model. The experiments were performed at a quasi-static strain rate Material mechanical response under torsion loadings with their associated variability.


Figure 16 Comparison of experimental tension, compression and torsion data with a microstructure-property model. The experiments were performed at a quasi-static strain rate Material mechanical response under (a) tension, (b) compression, and (c) torsion loadings with their associated variability.

Figure 17a Comparison of experimental measure material Bauschinger behavior with a microstructure-property model under tension followed by compression.
Figure 17b Comparison of experimental measure material Bauschinger behavior with a microstructure-property model under compression followed by tension with their variability.


Figure 17 Comparison of experimental measure material Bauschinger behavior with a microstructure-property model under (a) tension followed by compression, and (b) compression followed by tension with their variability.

Finally it should be noted that the results presented here include initial distributions and correlation effects of material mechanical responses and microstructural parameters. The use of other probability distributions may have a strong effect on the quantitative values of the results, while the qualitative values still hold.

4.2 Model Verification

For model verification, we performed simulations using ABAQUS finite element software under different loading conditions (tension, compression, torsion, tension followed by compression and compression followed by tension). Figures 18 and 19 show the stress-strain curves comparing the finite element simulations results to the results obtain through calibration processes at different stress states.

Figure 18a Comparison of finite element simulated mechanical responses with results obtained through calibration process under tension.
Figure 18c Comparison of finite element simulated mechanical responses with results obtained through calibration process under torsion.
Figure 18b Comparison of finite element simulated mechanical responses with results obtained through calibration process under compression .


Figure 18 Comparison of finite element simulated mechanical responses with results obtained through calibration process under (a) tension, (b) compression and (c) torsion.

Figure 19a Comparison of finite element simulated Bauschinger behavior with results obtained through calibration process under tension followed by compression.
Figure 19b Comparison of finite element simulated Bauschinger behavior with results obtained through calibration process under compression followed by tension.


Figure 19 Comparison of finite element simulated Bauschinger behavior with results obtained through calibration process under (a) tension followed by compression, and (b) compression followed by tension.

Figures 18 and 19 show good correlations and show how the model captured the differences between the work hardening rate in tension, compression, and torsion, which express the importance of a physically motivated internal state variable plasticity and damage continuum model for modeling microstructural details. The model is able to capture the history effects arising from the boundary conditions and load histories, the microstructural defects and progression of damage from these defects and microstructural features such as, second phase particles, and intermetallics.

4.3 Model Validation For the model validation, we conducted notched tensile tests as shown in Figure 20. Stress triaxiality is the primary driving factor for damage in porous materials. In a uniform material, the notch geometry induces a smooth stress triaxiality field with a maximum value near the center of the specimen. In a damaged medium, stress concentrations induced by the presence of pores, may cause local regions of high stress triaxiality. The notch tests were conducted with constant cross-head speed of 5 inches per minute. However, a MTS knife blade axial extensometer with a 1-inch gage length was used for the strain measurement and was set at a full scale of 25% strain and calibrated to better than 0.25% for the full scale range. The load cell was calibrated to within 0.25% error reading through the full-scale range. The data was collected on a System 5000 Data Acquisition system. Figure 21 shows a good comparison of notch specimen experiment data with finite element simulation result.


Figure 20 The 0.025 inch radius notched Bridgmen specimen made of AA7075-T651 shown with the 1-inch axial extensometer placed across the notch to measure displacement.


Figure 20 The 0.025 inch radius notched Bridgmen specimen made of AA7075-T651 shown with the 1-inch axial extensometer placed across the notch to measure displacement.

Figure 21 Comparison of experimental measured load-displacement response of aluminum alloy 7075-T651 with a calibrated microstructure-property model for a notch radius of 0.025 in.


Figure 21 Comparison of experimental measured load-displacement response of aluminum alloy 7075-T651 with a calibrated microstructure-property model for a notch radius of 0.025 in.

Concluding Remarks

The rational and motivation of this study is to understand the effect of material internal heterogeneity and boundary conditions on localized damage and its progression mechanism. We analyzed the effect of uncertainty in microstructural features (i.e., voids, cracks, inclusions) along with uncertainties in loading and boundary conditions on the mechanical response and damage evolution (or accumulation) of a wrought aluminum alloy. We first calibrated the microstructure-property relationship model with monotonic mechanical responses (tension, compression, torsion, tension followed by compression, and compression followed by tension) and material microstructure features. Based on the calibration results, we quantified the influence of uncertainty of various material model parameters in the constitutive equations on uncertainty in damage. During the calibration process, we have also shown that a physically motivated material model was necessary to understand and predict the stress states and damage states associated with rolled aluminum containing large intermetallics. The uncertainty distribution related to material parameters associated with the plasticity and microstructural features were also tabulated during calibration processes and shown that almost every material mechanical response and microstructural parameters have the same distribution shape (Gaussian) but different uncertainty level (the spread of distribution). Having calibrated the model with monotonic loadings, validation and verification simulations and tests were performed using different types of loading, geometrical, and boundary conditions. The modeling result show good correlation with experimental data. This study demonstrates a method of calibrating, verifying and validating a microstructure-property relationship based internal state variable form material model. However, this chapter’s important contribution is the approach of modeling and quantifying uncertainties associated with the damage from particles. Thus, this approach could be used to model composites and other heterogeneous materials. The authors believe that the good correlation of the internal state variable plasticity and damage continuum model to the experimental results along with sensitivity analysis support the physically-based modeling of void nucleation, growth, and coalescence resulting from large and small particles.

The following conclusions are drawn from the results of this work.

1. Better correlation with experimental compression, tension, and torsional stress-strain curve was demonstrated.

2. Uncertainty calculations revealed that the distribution of material damage, void nucleation, and growth changes related to the applied strain path. It also revealed that small variation in experimental data could lead to significant variations in strain to failure.

3. Uncertainty calculations revealed that the initial isotropic damage (symmetric) evolved into an anisotropic form (asymmetric) as applied strain is increased, consistence with experimentally observed behavior for 7075 aluminum alloy in literature.

4. The spread of damage evolution distribution increases with the applied strain even though the uncertainties of initial microstructure distributions remain the same.

5. The sensitivities of the uncertainty of damage to the uncertainties of the input material responses and microstructural parameters were found to be dependent on the strain values. As the strain value changed (that is, as the damage evolved), the importance of theses material responses and microstructural parameters changed.

6. The sensitivities were found to be consistent with the physics of the damage progression for this particular type of material. At the very beginning, the initial defect size and number density of cracked particles are important. As the damage evolves, more voids are nucleated, and grow. Finally, voids combine with each other and coalescence becomes the main driver. It also shows that the damage evolution equations provide an accurate representation of the damage progression due to large intermetallic particles.

7. Finally, we have shown that the initial variation in the microstructure clustering lead to about ±7.0%, ±8.1%, and ±9.75% variation in the elongation to failure strain for torsion, tension, and compression loading respectively.

References

  1. Olson, G.B., (1997), “Computational Design of Hierarchically Structured Materials,” Science, 277, 1237–1242.
  2. McDowell, D.L., (2005), “Design of Heterogeneous Materials,” Journal of Computer-Aided Materials Design, 11(2-3), 81-83
  3. 3.0 3.1 Coleman, H.W. and Steele, W.G., (1999), “Experimentation and Uncertainty Analysis for Engineers,” 2nd Edition, John Wiley, New York, NY.
  4. McDowell, D.L., Choi, H.J., Panchal, J., Austin, R., Allen, J., and Mistree, F., (2007), “Plasticity-Related Microstructure-Property Relations for Materials Design,” Key Engineering Materials, 340-341, 21-30
  5. 5.0 5.1 5.2 5.3 5.4 Horstemeyer, M.F., Solanki, K., and Steele, W.G., (2005), “Uncertainty Methodologies to Characterize a Damage Evolution Model,” Plasticity Conference
  6. Subbarayan, G., and Raj, R., (1999), “A Methodology for Integrating Materials Science with System Engineering”, Mater. Des., 20, 1-20.
  7. Garrison, W.M. Jr., and Moody, N.R., (1987), “Ductile Fracture,” J. of Physics and Chem. of Solids, v 48 (11), 1035-1074
  8. Tirosh, J., Shirizly, A., and Rubinski, L., (1999), “Evolution of Anisotropy in the Compliances of Porous Materials during Plastic Stretching or Rolling – Analysis and Experiments,” Mechanics of Materials, 31, 449–460.
  9. Gray, G.T., Vecchio, K.S., and Lopez, M.F., (2000), “Microstructural Anisotropy on the Quasi-Static and Dynamic Fracture of 1080 Eutectoid Steel,” In: Explomet Conference, Albuquerque, NM, June 19–22.
  10. Tvergaard, V., and Needleman, A., (1984), “Analysis of Cup-Cone Fracture in A Round Tensile Bar,” Acta Metall., 32, 157–169.
  11. Clayton, J.D., and McDowell, D.L., (2004) “A Homogenized Finite Elastoplasticity and Damage: Theory and Computations,” Mechanics of Materials, 36, 799–824.
  12. 12.0 12.1 12.2 12.3 12.4 Horstemeyer, M.F., Lathrop. J., Gokhale. A.M., and Dighe, M., (2000a), “Modeling Stress State Dependent Damage Evolution in A Cast Al-Si-Mg Aluminum Alloy,” Theory and Appl. Fract. Mech., 33, 31-47.
  13. Rice J.R., (1971), “Inelastic Constitutive Relations for Solids: An Internal-Variable Theory and Its Application to Metal Plasticity,” Journal of the Mechanics and Physics of Solids, 19 (6), 433-455.
  14. 14.0 14.1 14.2 14.3 Horstemeyer, M.F., and Gokhale, A.M., (1999), “A Void-Crack Nucleation Model for Ductile Metals,” Int. J. Solids and Structures, 36, 5029-5055
  15. 15.0 15.1 15.2 15.3 Horstemeyer, M.F. and Ramaswamy, S., (2000b), “On Factors Affecting Localization and Void Growth in Ductile Metals: A Parametric Study,” Int. J. Damage Mech., 9, 6-28.
  16. 16.0 16.1 16.2 16.3 Horstemeyer, M.F., (2001), “From Atoms to Autos A New Design Paradigm Using Microstructure-Property Modeling, Part 1: Monotonic Loading Conditions,” Sandia National Laboratories, Sand2000-8662, March 2001.
  17. 17.0 17.1 Bammann, D.J., (1990), “Modeling Temperature and Strain Rate Dependent Large of Metals,” Applied Mechanics Reviews, 43 (5), Part 2, May
  18. 18.0 18.1 18.2 Bammann, D.J., Chiesa, M.L., Horstemeyer, M.F., and Weingarten, L.I., (1993), “Failure in Ductile Materials Using Finite Element Methods,” eds. N. Jones and T. Weirzbicki, Structural Crashworthiness and Failure, Applied Science.
  19. Horstemeyer, M.F., and Revelli, V., (1996), “Stress History Dependent Localization and Failure Using Continuum Damage Mechanics Concepts,” ASTM Symposium the Applications of Continuum Damage Mechanics to Fatigue and Fracture, STP1315, ed. D. McDowell.
  20. Horstemeyer, M.F., and Wang, P., (2003c), ‘Cradle-to-Grave Simulation-Based Design Incorporating Multiscale Microstructure-Property Modeling: Reinvigorating Design with Science,” Journal of Computer-Aided Materials Design, 10, 13-34, 2003.
  21. Horstemeyer, M.F., Negrete, M., and Ramaswamy, S., (2003a), “Using a Micromechanical Finite Element Parametric Study to Motivate a Phenomenological Macroscale Model For Void/Crack Nucleation in Aluminum with a Hard Second Phase,” Mechanics of Materials, 35, 675-687.
  22. 22.0 22.1 Horstemeyer, M.F., Baskes, M.I., Prantil, V.C., Philliber, J., and Vonderheide, S., (2003b), “A Multiscale Analysis of Fixed-End Simple Shear Using Molecular Dynamics, Crystal Plasticity, and A Macroscopic Internal State Variable Theory,” Modeling Simul. Mater. Sci. Eng, 11, 265-286.
  23. Graham-Brady, L.L., Arwade, S.R., Corr, D.J., Gutierrez, M.A., Breysse, D., Grigoriu, M., and Zabaras, N., (2006), “Probability and Materials: from Nano- to Macro-Scale: A Summary,” Probabilistic Engineering Mechanics, 21, 193-199.
  24. Bammann, D.J., (1984), “An Internal Variable Model of Viscoplasticity,” eds. Aifantis, E. C. and Davison, L., Media with Microstructures and Wage Propagation, Int. J. Engineering Science, 8-10, 1041.
  25. Bammann, D.J., and Aifantis, E.C., (1989a), “A Model for Finite-Deformation Plasticity,” Acta Mechanica, 69, 97-117.
  26. Bammann, D.J., and Aifantis, E.C., (1989b), “A Damage Model for Ductile Metals,” Nuclear Engineering and Design, 116, 355-362.
  27. Bammann, D.J., Chiesa, M.L., and Johnson G.C., (1996), “Modeling Large Deformation and Failure in Manufacturing Processes,” App. Mech., eds, Tatsumi, Wannabe, Kambe. Elsevier Science: 256-276.
  28. 28.0 28.1 Coleman, H.W. and Steele, W.G., (1999), “Experimentation and Uncertainty Analysis for Engineers,” 2nd Edition, John Wiley, New York, NY.
  29. Dighe, M.D., Gokhale, A.M., and Horstemeyer, M.F., (2002), “Effect of Strain Rate on Damage Evolution in a Cast Al-Si-Mg Base Alloy,” Metall. Mat. Trans A, 33A, 555-565
  30. The MathWorks Inc., (2008), http://www.mathworks.com.
  31. Carino, R., Banicescu, I., and Gao, W., (2006). “Dynamic Load Balancing with MatlabMPI,” eds. V.N. Alexandrov, et al, Lecture Notes in Computer Science 3992, Springer: 430-437.


Citation: Solanki, K.N., Horstemeyer, M.F., Steele, G.W., Hammi, Y., & Jordon, J.B. (2010). Calibration, Validation, and Verification Including Uncertainty of a Physically Motivated Internal State Variable Plasticity and Damage Model, International Journal of Solids and Structures, 47(2), 186-203.

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