International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Efficient stochastic simulation of dynamic brittle strength using a random perturbation-based micromechanics… Graham-Brady, Lori L.; Liu, Junwei Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Efficient Stochastic Simulation of Dynamic Brittle Strength Using a Random Perturbation-based Micromechanics Model Lori L. Graham-Brady Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA Junwei Liu Graduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA ABSTRACT: Ceramic materials exhibit high strength against ballistic impact loads. The failure mechanism in this context is associated with crack growth and coalescence. The properties of pre-existing flaws at the micro-scale, including the size, shape, orientations and clustering, have profound impact on the strength of such materials. Since the properties of pre-existing flaws are highly heterogeneous in space, the strength exhibits strong spatial variations, leading to localization of stress and subsequent failure. One approach to simulation of this spatial variability is to assign samples of the random flaw population statistics (e.g., flaw density and flaw size distribution) to each integration point of a macro-scale finite element analysis. Here we propose an up-scaling technique based on the micro-mechanics model proposed by Paliwal and Ramesh (2008)1. While in concept it is possible to perform a micromechanical analysis at each integration point individually, this becomes computationally prohibitive for macro-scale models with many elements. Instead, we propose a more efficient approach that applies a Taylor series expansion approximation to the constitutive behavior, based on the results of a single reference analysis from the micromechanics model. The reference parameters are taken from analysis of a typical parameter set for the pre-existing flaws. Peak strength and the corresponding strain, along with some necessary gradient results are recorded. Monte Carlo simulation of the material performance is therefore achieved by generating the random variables that represent the flaw population at every integration point, which typically require much less computational effort than stochastic simulation of representative constitutive property fields. With this approach a large-scale statistical study can be performed with high efficiency, with a speed up of approximately 2 orders of magnitude, while the relative error is satisfyingly low.  1. INTRODUCTION The typical failure mode of brittle materials under dynamic load is associated with crack growth and coalescence, which presents highly nonlinear and load-dependent mechanical response. At the micro-scale, brittle materials contain numerous pre-existing flaws, which are the nucleation sites for micro crack development under applied loads. The pre-existing flaws can be voids, weak grain boundaries, second phases, crystal defects, soft or hard inclusions, etc. The properties of pre-existing flaws at the micro-scale, including the size, shape, orientations and clustering, have profound impact on the strength of such materials. Various micro-mechanical models have been developed that take into account the influence of micro flaws on the mechanical response of material on meso-scale, for instance the Paliwal and Ramesh [2008] model.  In the Paliwal and Ramesh [2008] model, the constitutive relationship between the stress   and strain   during the damage process is given by: 𝜎 = 𝐸? βˆ’ 𝐸?Ξ© πœ–  (1) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 where 𝐸? is the modulus of intact material, and Ξ©  is the damage measurement and is evaluated by:  Ξ© = πœ‚?𝑙???  (2) in which πœ‚? is the number density of the π‘˜th flaw family in the RVE, and 𝑙? is the size of crack associated with this family.  Since the population of pre-existing flaws is heterogeneously distributed in space, the strength exhibits significant spatial variation, leading to localization of stress and subsequent failure. While in concept it is possible to perform a full micromechanical analysis at each integration point of a larger-scale model, this becomes computationally prohibitive for macro-scale models with many elements. Instead, we propose a more efficient approach that applies a Taylor series expansion approximation to the constitutive behavior, based on the results of a single reference analysis from the micromechanics model. In what follows we explore such a method, which we called the β€œTransferring method”.  2. PREDICTION OF STRENGTH WITH HIGH EFFICIENCY We consider the micromechanical modeling for uniaxial loading conditions, with the assumptions of constant strain rate πœ– applied to the samples. In this case local flaw density is the only perturbed material property.  First consider a vector of reference flaw densities πœ‚? . A complete micro-mechanical modeling incorporating this set of flaw statistics data is performed, and is named the reference run. During modeling, the time step 𝑑?  when the strength is reached can be identified by 𝜎(𝑑?) = 0 . Differentiating the constitutive relation Eq. (1) with respect to time gives us the relations of the strain, damage and damage rate at this instant:  Ξ©?πœ– + Ξ©?πœ–? = ???? πœ– Β  (3)  Where Ξ©? = Ξ©(𝑑?) , Ξ©? = Ξ© Β (𝑑?) , πœ–? = πœ–(𝑑?) and the subscript 𝑝 denotes the parameters at the time step corresponding to the strength. Next consider a new set of flaw densities πœ‚?? , which deviate from the reference one. We denote the data and results obtained with this new flaw density by a prime note. The damage, damage rate and strain present new values at the instant at which the peak stress (or strength) occurs, and Eq. (3) is updated as:  Β (Ξ©? + ΔΩ?)πœ– + (Ξ©? + ΔΩ?)(πœ–? + Ξ”πœ–?) = ???? πœ–   (4) in which the Ξ” terms indicates the difference of the corresponding parameters to the reference ones. Since the strain rate πœ– is set to be constant, relations of the perturbed variables can be estimated by subtracting Eq. (4) from Eq. (3), and ignoring the higher order terms:   ΔΩ?πœ– +   ΔΩ?πœ–? + Ξ©?Ξ”πœ–? = 0 (5) In the above equation, ΔΩ?  and ΔΩ?  can be expressed as function with respect to Ξ”πœ–?  and gradients from the reference run. When the strength occurs, the damage parameters in the two modelings can be evaluated by, respectively:   Ξ©? = πœ‚?𝑙ℒ??  (6) Ξ©?? = βˆ‘πœ‚?? 𝑙ℒ + ????? ?? ΔΩ(𝑑?)+ ???β„’ ?? Ξ”πœ–? ? Β  Β    (7) Subtracting Ξ©?Ξ©?'  from Ξ©?? Ξ©? , reorganizing the result and ignoring higher order terms, we get:  ΔΩ? = ΔΩ? + 𝛼?Ξ”πœ–? (8) Similarly, ΔΩ? Β can also be evaluated by:  ΔΩ? = ΔΩ? + 𝛽?Ξ”πœ–? Β  Β  (9) In Eq. (8) and (9), ΔΩ? , 𝛼? , ΔΩ?  and 𝛽?  are functions with respect to the parameters Ξ”πœ‚? , 𝑙ℒ  and gradients such as  Β ????? ?? , ???β„’ ?? , which are known or can be evaluated in the reference run. The perturbation in the strain at time tp is: 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3  Ξ”πœ–? = βˆ’ ????????????????????? Β  (10) Ξ”πœ–? is easily obtained by substituting the stored data and the perturbing flaw densities into the above equations, and then ΔΩ? and the strength 𝜎??  are solved in sequence through Eq. (8) and (1). 3. RESULTS  Using both brute-force Monte Carlo simulations and the proposed transferring method, the peak stress under uniaxial compressive loading was predicted and is shown in Figure 1. The input material properties for this analysis are elastic modulus  𝐸 = 300GPa, Poisson’s ratio 𝜈 = 0.24, mean areal flaw density  πœ‚ = 10?/π‘š?, variance of flaw density VAR πœ‚ = 2.5Γ—10β„’ , mean flaw size 𝑠 = 26¡μm, flaw orientation πœ™ = 50.7∘, and strain rate πœ– = 10?𝑠??. The strengths predicted from the proposed method (denoted by blue circles in Figure 1) agree very well with those obtained using Monte Carlo simulations (denoted by green stars in Figure 1), with only slight errors at the extreme values of the distribution. The transferring method required only 10?? of the computational time required from the direct Monte Carlo simulations. 4. CONCLUSIONS AND DISCUSSION Using the approximating methodology referred to here as transferring we can directly evaluate strength given altered statistics of the flaw population, without re-running the complete micro-mechanical model. The accuracy of this approximation is typically excellent, although the level of accuracy depends on the difference between the new statistical data and the data in the reference runs. With the perturbation in the statistical realizations of the flaws and the reference data readily generated, computing the entire range of strength with this methodology only takes a small fraction of the time required by multiple runs of the micromechanical model. Future work will focus on implementing the transferring method in a stochastic finite element context, in which local material properties are evaluated at each integration point based on stochastic variations in the microstructural characterization. This enables the development of physically-based stochastic finite elements.     Figure 1: Uniaxial compressive stress vs. strain at a strain rate of 10?𝑠?? , as predicted by the micromechanics model. Peak stress predictions from the proposed model (blue circles) agree very well with those from the explicit Monte Carlo-based approach (green stars), with an increase of efficient of 10?.  5. ACKNOWLEDGEMENTS The authors gratefully acknowledge support of the US Army Research Laboratory. The research was accomplished under Cooperative Agreement Number W911NF-12-2-0022. 6. REFERENCES Paliwal, B., Ramesh, K.T. (2008). An interacting micro-crack damage model for failure of brittle materials under compression, Journal of the Mechanics & Physics of Solids 56, 896-923.  

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