1 Introduction
In order to predict the behavior of mechanically loaded metallic materials, constitutive models are applied, which present a mathematical frame for the description of elastic and inelastic deformation. The models by Miller, Krempl, Korhonen, Aubertin, Chan, and Bodner are wellknown constitutive models for isotropic materials [1, 2, 3, 4, 5] which describe the inelastic behavior. In 1983, Chaboche [6, 7] put forward what has become known as the unified Chaboche viscoplasticity constitutive model, which has been widely accepted.
All inelastic constitutive models contain parameters which have to be identified for a given material from experiments. In the literature only few investigations can be found dealing with identification problems using stochastic approaches. Klosowski and Mleczek have applied the leastsquares method in the MarquardtLevenberg variant to estimate the parameters of an inelastic model [8]. Gong et al. have also used some modification of the leastsquares method to identify the parameters [9]. Harth and Lehn identified the model parameters of a model by employing some generated artificial data instead of experimental data using a stochastic technique [10]. A similar study by Harth and Lehn has been done for other constitutive models like Lindholm and Chan [11].
There are few investigations on the simplest material model, the elasticity model, to identify only very few parameters of the model where sampling approaches like the Metropolis algorithm and its modifications are employed. Pacheco et al. [12] investigated a threepoint bending test to identify elastic behavior and calibration is done by solving the inverse problem in a Bayesian setting by using the MetropolisHastings algorithm. Only elastic moduli are identified without considering the error. Slonski et al. [18] also applied a sequential particle filter on an elastic model and Young’s modulus is estimated, where the Bayesian setting is compared to the deterministic approach, and the Bayesian setting is preferred. The elastic modulus of a polymeric material is updated by Zhang et al. [24], where a Markov Chain Monte Carlo method is used, but very high computation time is reported. Further, Young’s modulus is estimated for a considered material by Gallina et al. [13] by applying a multi dimensional Markov Chain Monte Carlo method. A similar Bayesian approach for composite materials to estimate Young’s modulus is carried out by Pieczonka et al. [14]. Arnst et al. [15] have applied a Markov Chain Monte Carlo method by using polynomial chaos expansion to identify Young’s modulus of an elastic model.
Only few investigations were carried out on enriched material models such as a viscoelastic model to identify the few parameters by employing the MetropolisHastings technique and the classical Markov Chain Monte Carlo method. Rappel et al. [16, 17]
studied an elastic and a viscoelastic model where the measurement error is considered. Bayesian inference is applied to estimate only the elasticity modulus by applying an adaptive MetropolisHastings technique. An et al.
[19] investigated a crack model by a classical Markov Chain Monte Carlo method in order to estimate the parameters of a model which represent the size and position of the crack in the Bayesian setting. Also Hernandez et al. [20] applied a Markov Chain Monte Carlo method on a viscoelastic model in order to update its model parameters in a Bayesian setting, but the posterior distributions of the parameters are not updated properly. In fact, the parameters are not identified properly. Mahnken [21] has also applied a Markov Chain Monte Carlo method to estimate few parameters of a plasticity model. The damage parameters of a truss structure under model uncertainties are studied by Zheng et al. [22], where a multilevel Markov Chain Monte Carlo method is applied and the true values are not well estimated. Further, this approach suffers from high computation time. Another damage detection approach is applied by Nichols et al. [23] by applying a modified version of the Markov Chain Monte Carlo method.There are other investigations in the Bayesian setting using the Markov Chain Monte Carlo method, and Madireddy et al [25], Wang and Zabaras [26], and Oh et al [27] have carried out an investigation on the identification of a material model by using its modified method. In studies carried out by Alvin [28], Marwala and Sibusiso [29], Daghia et al. [30], Abhinav and Manohar [31], Gogu et al. [32, 33], and Koutsourelakis [34, 35], the elastic parameters of the model are estimated stochastically. Fitzenz et al. [36], Most [37], and Sarkar et al. [38] investigated elastoplastic materials and thermodynamical material models to identify their model parameters. Other studies on viscoelastic models are carried out by Zhang et al. [39], Mehrez et al. [40], and Miles et al. [41]. Further investigations on viscoelastic models to estimate a higher number of model parameters are studied by Zhao and Pelegri [42], and by Kenz et al. [43]. The estimation of fatigue parameters using Markov Chain Monte Carlo methods is also studied by [44].
In this paper, a viscoplastic model of Chaboche is studied. The model contains five material parameters which have to be determined from experimental data. It should be noted that here virtual data are employed instead of real experimental data. A cyclic tensioncompression test is applied in order to extract the virtual data.
The model is described in Section 2, whereas Section 3 explains how to perform the update. The employed method, which is the Transitional Markov Chain Monte Carlo (TMCMC) method, is described in this section and applied on the case study model.
In Section 4 the desired parameters of the model are identified from the measured data which .. take as surface displacement measurements, as could be done for example by digital image correlation (DIC). In fact, the parameters which have been considered as uncertain parameters are updated and the uncertainties of them are reduced, while the random variables representing the uncertain parameters are updated during the process. The results are thoroughly studied and the identified parameters as well as the corresponding model responses are analyzed. The section discusses the ability of the applied method to identify the highly nonlinear viscoplastic model parameters by considering the surface displacement measurement.
2 Model Problem
The mathematical description of metals under cyclic loading beyond the yield limit that includes viscoplastic material behavior as well as the characterization of isotropickinematic hardening is here given in terms of a modified Chaboche model introduced in [45]
. As we consider classical infinitesimal strains, we assume an additive strain decomposition of elastic and inelastic strains. The material behavior is described for the elastic part by isotropic homogeneous elasticity which is characterized by the shear and bulk moduli, the eigenvalues of the elasticity tensor. The inelastic behavior is described following the standard generalized materials paradigm
[46], so that obey the specification of the dissipation pseudopotential is necessary. For viscoplasticity the dissipation potential is given by(1) 
with and and as the material parameters. Here is the overstress, defined via the equivalent stress () which reads
(2) 
where denotes the deviatoric part and is the backstress of kinematic hardening. The overstress is given by
(3) 
where is the yield stress and models the isotropic hardening which is introduced in the following. The partial derivative of the dissipation potential with respect to leads to the equation for the inelastic strain rate
(4) 
The viscoplastic model allows for isotropic and kinematic hardening, which is considered in order to describe different specifications. Assuming and with and to describe isotropic and kinematic hardening respectively, the evolution equations for these two are
(5) 
and
(6) 
respectively. In the evolution equations, is the viscoplastic multiplier rate given as:
(7) 
which describes the rate of accumulated plastic strains. The parameter indicates the speed of stabilization, whereas the value of the parameter is an asymptotic value according to the evolution of the isotropic hardening. Similarly, the parameter denotes the speed of saturation and the parameter is the asymptotic value of the kinematic hardening variables. The complete model is stated in Table 1. Note that represents the elasticity tensor.
By gathering all the desired material parameters to identify into the vector , where and are bulk and shear modulus, respectively, which determine the isotropic elasticity tensor, the goal is to estimate given measurement displacement data, i.e.
(8) 
in which represents the measurement operator giving the prediction of the observed displacements and is the measurement (also possibly the model) error. As the function is typically not invertible, this is an illposed problem, and the estimation of given is not an easy task and usually requires regularization. This can be achieved either in a deterministic or in a probabilistic setting. Here, the latter path is taken as further described in the text.
3 Bayesian identification via the Transitional Markov Chain Monte Carlo method
Ching and Chen [48] modified the approach by Beck and Au to develop a Transitional Markov Chain Monte Carlo method (TMCMC). This approach is based on the Markov Chain Monte Carlo method and its adaptive capability is inspired from an adaptive Metropolis–Hastings (MH) method developed by Beck and Au in 2002 [47]. This approach employs a sequence of PDFs, where the entire data set of the available data is used for each stage of the sampler which is proportional to the local posterior with some exponent times likelihood, with a change of exponent between 0 and 1. As the same idea was used in the simulated annealing approach [49], this exponent is called a tempering parameter. TMCMC and the Beck and Au approach also differs in theory of employing the MH algorithm. In each stage a local proposed PDF is constructed instead of a global proposed PDF, and it is also to be noted that to improve the rate of convergence in TMCMC, resampling is used.
Based on the formulation of Bayes’s theorem [50, 51]
, it is difficult to generate samples from the posterior PDF due to lack of information about the geometry of the probability density function. In order to overcome this problem, the TMCMC algorithm employs a sequence of intermediate PDFs which converge to the target posterior PDF as defined in the Equation
9, where the index denotes the stage number [48, 52]. Considering the reformulation of Bayes’s theorem for the model class which is defined by the parameter vector , the prior PDF of this model class over the parameter vector is updated with the measurement data , where the likelihood function is .(9a)  
(9b) 
The stage number fulfills the desirable properties such as producing a series of intermediate PDFs () from the prior PDF, i.e. it starts with stating that the initial prior is proportional to the prior PDF () and it ends with stating that the final PDF is proportional to the posterior PDF (). The right hand side of the latter relation represents the posterior PDF and its proportionality is shown in Equation 10.
(10a)  
(10b) 
Although the geometry change from to is large, the change between two adjacent intermediate PDFs can be small. Due to this possible transition, from where the method derives its name, it is possible to determine samples efficiently from based on samples from .
3.1 Transitional MCMC Algorithm
In this subsection, the entire algorithm of TMCMC is described in the following steps in detail [48, 52, 53].

As a first step, set , and generate from a set of samples.

It is to be noted that is selected in such a way that the coefficient of variation of where is a prescribed value and stands for the th sample that belongs to level . Accordingly, the coefficient of variation serves as an indicator to measure the closeness of to .

The plausibility weight is obtained for , from which the parameter is computed.

Samples where determined from by MetropolisHastings technique are chosen randomly from a Markov chain which has samples starting from one of the samples where . The ith initial sample is chosen with probability
. As in the proposed distribution a Gaussian distribution which is centered at the current sample is applied in the kth chain in the MetropolisHastings algorithm, and the determined covariance matrix is shown in Equation
11 and 12, where is a prescribed scaling factor that scales the proposal distribution and is suggested by Ching and Chen [48].
(11) where
(12)

Steps two to four are repeated until . At the final step, samples for are distributed according to .
The significant advantage of the modified MetropolisHastings algorithm applied in TMCMC is that initially it allows a large and a free sample space to explore but in the final stages the samples are generated from a narrower neighborhood of sample space. Furthermore the proposal distribution will change within a simulation level and this will result in better local behavior. This is accomplished by modifying the proposal distribution for each level in such a way that its standard deviation is small for higher simulation levels. But the average value drives the generation of samples towards the most important neighborhood of the sample space
[48, 52, 53].4 Numerical results
The identification of the material constants in the Chaboche unified viscoplasticity model is a reverse process, here based on virtual data. In case of the Chaboche model the best way of parameter identification is using the results of cyclic tests, since more information can be obtained from cyclic test rather than creep and relaxation tests, specifically information regarding hardening parameters. The aim of the parameter identification is to find a parameter vector introduced in Section 2. The bulk modulus (), the shear modulus (), the isotropic hardening coefficient (), the kinematic hardening coefficient () and the yield stress () are considered as the uncertain parameters of the constitutive model.
A preliminary study is on a regular cube, modeled with one 8 node element. The minimal number of freedoms that have to be constrained is six and many combinations are possible. As shown in Figure 1
, all three degrees of freedom at point B are fixed. This prevents all rigid body translations, and leaves three rotations to be taken care of. The
displacement component at point A is constrained to prevent rotation about , and the component is fixed at point C to prevent rotation about . The component is constrained at point D to prevent rotation about [54].The normal tractions, which is a Neumann boundary condition, are applied cyclically [55, 56, 57] in , and directions on front and back faces and the magnitude of tractions in all directions are shown in Figure 2 where green, red and blue colors represent the stress vector values in , and directions respectively.
By considering the parameters listed in Table 2 as the virtual truth, for the top right corner node on back face, point E, as shown in Figure 1, the related displacement graph is obtained as shown in Figure 3 where green, red and blue colors represent the displacement of point E in , and directions respectively.

The displacements of point E in , and directions are noted as the virtual data in this case.
The Transitional Markov chain Monte Carlo method is applied using sampling by which 1000 samples are generated and the history of the displacement of point E is noted. The determined prior and posterior probability density functions are compared and illustrated in Figure
5 and 5. It should be noted that the Burnin period is considered as in all iterations but in last iteration it is considered as . Further, the scaling parameter () in TMCMC is considered as [48]. The acceptance ratio is implemented in such an adaptive way that it is between 0.2 and 0.3, considering the optimal acceptance ratio 0.234 [58]. The initial prior of the material model parameters are taken as uniform distributions.From the prior and posterior distributions of the bulk modulus (), shear modulus (), the isotropic hardening coefficient (), the kinematic hardening coefficient () and the yield stress (), it is found that the model parameters can be detected by using the TMCMC. The obtained results are acceptable, however it should be mentioned that the TMCMC method is a very time consuming approach, particularly when the case study is a PDE system. The determined true values, the mean and standard deviation of the estimated parameters via the TMCMC approach are shown in Table 3.
Parameters  

50  53.00  1.83  
50  49.30  1.32  
4.1 Discussion of the results
From the sharpness of the posterior PDF of , , , and , it can be concluded that enough information from virtual data, the surface displacement measurement, is received, and updating the parameters considering their uncertainty is done very properly, as the standard deviation of the residual uncertainty is below of the mean. On the other hand, it should be mentioned that the Transitional Markov Chain Monte Carlo approach, which is employed, is a very time consuming method and applying it, besides a model which is solved by employing the finite element method as a relatively expensive method, leads to a computationally very expensive approach to estimate the model parameters.
5 Summary
Using the Transitional Markov Chain Monte Carlo method explained in Section 3 to identify the model parameters of the Chaboche model indicates that it is possible to identify the model parameters by employing this method. The method works well on this highly nonlinear material model by taking the vector of surface displacement measurement into consideration. The parameters are well estimated, and the uncertainty of the parameters is reduced, while the distribution functions of the parameters are updated during the process. The adaptive procedure of the acceptance ratio, where the acceptance ratio is accepted when it is between 0.2 and 0.3, leads to very accurate posterior distributions, which estimate the parameters very finely. As applying the TMCMC on the discussed material model needs a lot of computational time, a new much faster approach will be introduced to estimate the model parameters using the surface displacement measurement and the results will be compared on the same material model in the upcoming paper of the authors which can be proposed not only for the purpose of model parameter identification but also for the purpose of structural health monitoring and damage detection of the real structures such as bridges under cyclic tests.
Acknowledgement
This work is partially supported by the DFG through GRK 2075.
References
 [1] A. Miller. An Inelastic Constitutive Model for Monotonic, Cyclic, and Creep Deformation: Part I–Equations Development and Analytical Procedures. J. Eng. Mater. Technol. 98(2), 97105, 1976.
 [2] E. Krempl, J. J. McMahon, and D. Yao. Viscoplasticity Based on Overstress with a Differential Growth Law for the Equilibrium Stress. Mechanics of Materials 5, 3548, 1986.
 [3] R. K. Korhonen, M. S. Laasanen, J. Toyras, R. Lappalainen, H. J. Helminen, and J. S. Jurvelin. Fibril reinforced poroelastic model predicts specifically mechanical behavior of normal, proteoglycan depleted and collagen degraded articular cartilage. J Biomech 36, 13731379.
 [4] Michel Aubertin, Denis E. Gill, and Branko Ladanyi. A unified viscoplastic model for the inelastic flow of alkali halides. Mechanics of Materials 11, 638, 1991.
 [5] K. S. Chan, S. R. Bodner, A.F. Fossum, and D.E. Munson. A constitutive model for inelastic flow and damage evolution in solids under triaxial compression. Mechanics of Materials 14, 114, 1992.
 [6] J. L. Chaboche, and G. Rousselier. On the plastic and viscoplastic constitutive equations  part 1: rules developed with internal variable concept. J. Press. Vessel Technol., 105, 153158, 1983.
 [7] J. L. Chaboche, and G. Rousselier. On the plastic and viscoplastic constitutive equations  part 2: application of internal variable concepts to the 316 stainless steel. J. Press. Vessel Technol., 105, 159164, 1983.
 [8] P. Kłosowski, and A. Mleczek. Parameters’ Identification of Perzyna and Chaboche Viscoplastic Models for Aluminum Alloy at Temperature of . Engng. Trans. 62, 3, 291305, 2014.
 [9] Y. Gong, C. Hyde, W. Sun, and T. Hyde. Determination of material properties in the Chaboche unified viscoplasticity model. Journal of Materials Design and Applications, 224(1), 1929, 2010.
 [10] T. Harth, and Jürgen Lehn. Identification of Material Parameters for Inelastic Constitutive Models Using Stochastic Methods. GAMMMitt. 30, No. 2, 409429, 2007.
 [11] K. S. Chan, S. R. Bodner, and U. S. Lindholm. Phenomenological Modelling of Hardening and Thermal Recovery in Metals. Journal of Engineering Materials and Technology 110, 18, 1988.

[12]
C. Pacheco, G. S. Dulikravich, M. Vesenjak, M. Borovinsek, I. M. A. Duarte, R. Jha, S. R. Reddy, H. R. B. Orlande and M. J. Colaco. Inverse parameter identification in solid mechanics using Bayesian statistics, response surfaces and minimization. Technische Mechanik 36(12), 120131, 2016.
 [13] A. Gallina, L. Ambrozinski, P. Packo, L. Pieczonka, T. Uhl and W. J. Staszewski. Bayesian parameter identification of orthotropic composite materials using lamb waves dispersion curves measurement. Journal of Vibration and Control 23(16), 26562671, 2017.
 [14] L. Pieczonka, A. Gallina, L. Ambrozinski, P. Packo, T. Uhl and W. J. Staszewski. Parameters identification of composite materials using Bayesian approach and guided ultrasonic waves. Proceedings of ISMA 2016  International Conference on Noise and Vibration Engineering and USD2016, 2016.
 [15] M. Arnst, R. Ghanem and C. Soize. Identification of Bayesian posteriors for coefficients of chaos expansions. Journal of Computational Physics 229(9), 31343154, 2010.
 [16] H. Rappel, L. A. A. Beex, J. S. Hale and S. P. A. Bordas. Bayesian inference for the stochastic identification of elastoplastic material parameters: introduction, misconceptions and insights. arXiv:1606.02422, 2017.

[17]
H. Rappel, L. A. A. Beex, L. Noels and S. P. A. Bordas. Identifying elastoplastic parameters with Bayes’ theorem considering double error sources and model uncertainty. Journal of Probabilistic Engineering Mechanics, 2018.
 [18] M. Słonski. Bayesian identification of elastic parameters in composite laminates applying lamb waves monitoring. Computational Methods in Structural Dynamics and Earthquake Engineering, M. Papadrakakis, V. Papadopoulos, V. Plevris (Eds.), 2015.
 [19] D. An, J. Cho and N. H. Kim. Identification of correlated damage parameters under noise and bias using Bayesian inference. Structural Health Monitoring 11(3), 293303, 2011.
 [20] W. P. Hernandez, F. C. L. Borges, D. A. Castello, N. Roitman and C. Magluta. Bayesian inference applied on model calibration of a fractional derivative viscoelastic model. Proceedings of the XVII International Symposium on Dynamic Problems of Mechanics, V. Steffen, D. A. Rade, W. M. Bessa (Eds.), 2015.
 [21] R. Mahnken. Identification of material parameters for constitutive equations. Encyclopedia of Computational Mechanics Second Edition, Part 2. Solids and Structures, 121, 2017.
 [22] W. Zheng and Y. Yu. Bayesian probabilistic framework for damage identification of steel truss bridges under joint uncertainties. Advances in Civil Engineering 113, 2013.
 [23] J. M. Nichols, E. Z. Moore and K. D. Murphy. Bayesian identification of a cracked plate using a populationbased Markov Chain Monte Carlo method. Journal of Computers and Structures 89(1314), 13231332, 2011.
 [24] E. Zhang, J. D. Chazot and J. Antoni. Parametric identification of elastic modulus of polymeric material in laminated glasses. 16th IFAC Symposium on System Identification, The International Federation of Automatic Control, 2012.
 [25] S. Madireddy, B. Sista and K. Vemaganti. A Bayesian approach to selecting hyperelastic constitutive models of soft tissue. Journal of Computational and Applied Mathematics 291, 102122, 2015.
 [26] J. Wang and N. Zabaras. A Bayesian inference approach to the inverse heat conduction problem. International Journal of Heat and Mass Transfer 47(1718), 39273941, 2004.
 [27] C. K. Oh, J. L. Beck and M. Yamada. Bayesian learning using automatic relevance determination prior with an application to earthquake early warning. Journal of Engineering Mechanics 134(12), 10131020, 2008.
 [28] K. Alvin. Finite element model update via Bayesian estimation and minimization of dynamic residuals. The American Institute of Aeronautics and Astronautics Journal 135(5), 879886, 1997.
 [29] T. Marwala and S. Sibusiso. Finite element model updating using Bayesian framework and modal properties. Journal of Aircraft 42(1), 275278, 2005.
 [30] F. Daghia, S. de Miranda, F. Ubertini and E. Viola. Estimation of elastic constants of thick laminated plates within a Bayesian framework. Journal of Composite Structures 80(3), 461473, 2007.
 [31] S. Abhinav and C. Manohar. Bayesian parameter identification in dynamic state space models using modified measurement equations. International Journal of NonLinear Mechanics 71, 89103, 2015.
 [32] C. Gogu, W. Yin, R. Haftka, P. Ifju, J. Molimard, R. le Riche and A. Vautrin. Bayesian identification of elastic constants in multidirectional laminate from Moire interferometry displacement fields. Journal of Experimental Mechanics 53(4), 635648, 2013.
 [33] C. Gogu, R. Haftka, J. Molimard and A. Vautrin. Introduction to the Bayesian approach applied to elastic constants identification. The American Institute of Aeronautics and Astronautics Journal 48(5), 893903, 2010.
 [34] P. S. Koutsourelakis. A novel Bayesian strategy for the identification of spatially varying material properties and model validation: an application to static elastography. International Journal for Numerical Methods in Engineering 91(3), 249268, 2012.
 [35] P. S. Koutsourelakis. A multiresolution, nonparametric, Bayesian framework for identification of spatiallyvarying model parameters. Journal of Computational Physics 228(17), 61846211, 2009.
 [36] D. D. Fitzenz, A. Jalobeanu and S. H. Hickman. Integrating laboratory creep compaction data with numerical fault models: a Bayesian framework. Journal of Geophysical Research: Solid Earth 112, B08410, 2007.
 [37] T. Most. Identification of the parameters of complex constitutive models: least squares minimization vs. Bayesian updating. In: Straub, D. (ed.) Reliability and Optimization of Structural Systems, CRC Press, 119130, 2010.
 [38] S. Sarkar, D. S. Kosson, S. Mahadevan, J. C. L. Meeussen, H. van der Sloot, J. R. Arnold, K. G. Brown. Bayesian calibration of thermodynamic parameters for geochemical speciation modeling of cementitious materials. Journal of Cement and Concrete Research 42(7), 889902, 2012.
 [39] E. Zhang, J. D. Chazot, J. Antoni and M. Hamdi. Bayesian characterization of Young’s modulus of viscoelastic materials in laminated structures. Journal of Sound and Vibration 332(16), 36543666, 2013.
 [40] L. Mehrez, E. Kassem, E. Masad and D. Little. Stochastic identification of linearviscoelastic models of aged and unaged asphalt mixtures. Journal of Materials in Civil Engineering 27(4), 04014149, 2015.
 [41] P. Miles, M. Hays, R. Smith and W. Oates. Bayesian uncertainty analysis of finite deformation viscoelasticity. Journal of Mechanics of Materials 91, 3549, 2015.
 [42] X. Zhao and A. A. Pelegri. A Bayesian approach for characterization of soft tissue viscoelasticity in acoustic radiation force imaging. International Journal for Numerical Methods in Biomedical Engineering 32(4), E02741, 2016.
 [43] Z. R. Kenz, H. T. Banks and R. C. Smith. Comparison of frequentist and Bayesian confidence analysis methods on a viscoelastic stenosis model. SIAM/ASA Journal on Uncertainty Quantification 1(1), 348369, 2013.
 [44] D. An, J. Choi, N. H. Kim and S. Pattabhiraman. Fatigue life prediction based on Bayesian approach to incorporate field data into probability model. Journal of Structural Engineering and Mechanics 37(4), 427442, 2011.
 [45] J. Velde. 3D Nonlocal Damage Modeling for Steel Structures under Earthquake Loading. Department of Architecture, Civil Engineering and Environmental Sciences University of Braunschweig  Institute of Technology, 2010.
 [46] Q. S. Nguyen. Mechanical Modeling of Anelasticity. Revue de Physique Appliquee 23(4), 325330, 1988.
 [47] J. L. Beck and S. K. Au. Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation. Journal of Engineering Mechanics 128(4), 380391, 2002.
 [48] J. Ching and Y. C. Chen. Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection and model averaging. Journal of Engineering Mechanics 133(7), 816832, 2007.
 [49] G. S. Fishman. Coordinate selection rules for Gibbs sampling. University of North Carolina, UNC/OR TR92/15, 1996.
 [50] R. C. Aster, B. Borchers and C. H. Thurber. Parameter Estimation and Inverse Problems. 2nd Edition, Academic Press, 2012.
 [51] M. Dashti and A. M. Stuart. The Bayesian Approach to Inverse Problems. Handbook of Uncertainty Quantification, 1118. , 2015.
 [52] J. Ching and J. Wang. Application of the Transitional Markov Chain Monte Carlo algorithm to probabilistic site characterization. Journal of Engineering Geology 203, 151167, 2016.
 [53] J. Wang, L. S. Katafygiotis and M. N. Noori. Transitional Markov Chain Monte Carlo simulation for reliabilitybased optimization. Safety, Reliability, Risk and LifeCycle Performance of Structures Infrastructures – Deodatis, Ellingwood Frangopol (Eds), 15931599, 2014.
 [54] C. Felippa. Introduction to FEM, FEM Modeling: Mesh, Loads and BCs. Lecture Notes, University of Colorado Boulder, 2016.
 [55] E. Adeli, B. V. Rosić, H. G. Matthies and S. Reinstädler. Bayesian parameter identification in plasticity. XIV International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIV E. Oñate, D.R.J. Owen, D. Peric and M. Chiumenti (Eds), 2017.
 [56] E. Adeli, B. V. Rosić, H. G. Matthies and S. Reinstädler. Effect of Load Path on Parameter Identification for Plasticity Models using Bayesian Methods. Quantification of Uncertainty: Improving Effieciency and Technology, G. Rozza, M. D’Elia and M. Gunzburger (Eds), http://arxiv.org/abs/1906.07246, 2018.
 [57] E. Adeli. ViscoplasticDamage Model Parameter Identification via Bayesian Methods. Ph.D. Dissertation, Institut für Wissenschaftliches Rechnen, Technische Universität Braunschweig, 2019.
 [58] A. Gelman, G. O. Roberts and W. R. Gilks. Efficient Metropolis Jumping Rules. In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith (ed.), Bayesian Statistics 5, Oxford University Press, 599607, 1996.
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