A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials

Suzuki, Jorge L.; Zhou, Yongtao; D’Elia, Marta; Zayernouri, Mohsen

doi:10.1016/j.cma.2020.113494

Cited by 32 publications

(23 citation statements)

References 48 publications

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“…• Fractional calculus can be a interesting modeling alternative to describe the linear/nonlinear behavior of porcine UB especially within a material model selection framework, since fractional models potentially provide a reduced number of material parameters due to the presence of multiple power-laws in relaxation. Regarding potential future steps, investigating the possibility of plastic deformations under large strains by employing quasi-linear visco-plastic effects [61,62,63] and also failure mechanisms [7] would be interesting studies towards the life-cycle prediction of such anomalous bio-tissues. Finally the variation of tissue properties by anatomical location, orientation, and layers of the UB motivates further studies on distributed order models incorporating nonlinearities.…”

Section: Discussionmentioning

confidence: 99%

“…The corresponding rheological symbol for the SB model represents a fractal-like arrangement of springs and dashpots [56,40], which we interpret as a compact, upscaled representation of a fractal-like microstructure. Regarding the thermodynamic admissibility of the SB element and more complex models (i.e., plasticity and damage) involving it, we refer the reader to Lion [36] for the SB model, and Suzuki et al [63]. The relaxation function G SB (t) [P a] for the SB model is given by the following single, inverse power-law form:…”

Section: First Stage: An Existence Study Of Fractional Linear Viscoel...mentioning

confidence: 99%

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A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

Suzuki¹,

Tuttle²,

Roccabianca³

et al. 2021

Preprint

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We introduce a data-driven fractional modeling framework for linear and nonlinear viscoelasticity aimed at complex materials, and particularly bio-tissues. From consecutive uniaxial multi-step relaxation experiments (up to 200% strains) of five distinct anatomical locations of porcine urinary bladder, we identify an anomalous relaxation character, with two power-law-like behaviors for short and long times, and strain nonlinearity for applied step strains greater than 25%. Such behavior is affected by heterogeneous regional material compositions and nonlinearities taking place due to large strains. The first component of our modeling framework is an existence study, to determine admissible candidate fractional linear viscoelastic models, that qualitatively describe the (linear) time-dependent relaxation behavior. After the appropriate linear viscoelastic model is selected, the second stage include large-strain effects to the framework. For this purpose, among existing approaches in the literature, we utilize a fractional quasi-linear viscoelastic approach, given by a multiplicative kernel decomposition of the relaxation function from the selected linear model and a nonlinear elastic response for the bio-tissue of interest. From single-relaxation data of the urinary bladder, a fractional linear Maxwell model captures both short/long-term behaviors with two fractional orders, therefore being the most suitable candidate model for small strains at the first stage. For the second stage, multi-step relaxation data under large strains were employed to calibrate a four-parameter fractional quasi-linear viscoelastic model, that combines a Scott-Blair relaxation function and an exponential instantaneous stress response, the latter form being sufficient to describe the elastin/collagen phases of bladder rheology. Our obtained results demonstrate that the employed fractional quasi-linear model, with a single fractional order in the range α = 0.25 − 0.30 for distinct bladder samples, is suitable for the porcine urinary bladder, producing root mean squared errors below 2% without need for recalibration over subsequent applied step strains. Our analyzes demonstrate that fractional models arise as attractive tools to capture the bladder tissue behavior under small-to-large strains and multiple time scales, therefore being potential alternatives to describe multiple stages of bladder functionality.

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Section: Discussionmentioning

confidence: 99%

Section: First Stage: An Existence Study Of Fractional Linear Viscoel...mentioning

confidence: 99%

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

Suzuki¹,

Tuttle²,

Roccabianca³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The corresponding rheological symbol for the SB model represents a fractal-like arrangement of springs and dashpots [7,56], which we interpret as a compact, upscaled representation of a fractal-like microstructure. Regarding the thermodynamic admissibility of the SB element and more complex models (i.e., plasticity and damage) involving it, we refer the reader to Lion [57] for the SB model, and Suzuki et al [58]. The relaxation function G SB (t) [Pa] for the SB model is given by the following single, inverse power-law form:…”

Section: First Stage: An Existence Study Of Fractional Linear Viscoel...mentioning

confidence: 99%

“…Regarding potential future steps, investigating the possibility of plastic deformations under large strains by employing quasi-linear visco-plastic effects [58,68,77] and also failure mechanisms [78] would be interesting studies towards the life-cycle prediction of such anomalous bio-tissues. Finally the variation of tissue properties by anatomical location, orientation, and layers of the UB motivates further studies on distributed order models incorporating nonlinearities.…”

mentioning

confidence: 99%

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

et al. 2021

Self Cite

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We introduce a data-driven fractional modeling framework for complex materials, and particularly bio-tissues. From multi-step relaxation experiments of distinct anatomical locations of porcine urinary bladder, we identify an anomalous relaxation character, with two power-law-like behaviors for short/long long times, and nonlinearity for strains greater than 25%. The first component of our framework is an existence study, to determine admissible fractional viscoelastic models that qualitatively describe linear relaxation. After the linear viscoelastic model is selected, the second stage adds large-strain effects to the framework through a fractional quasi-linear viscoelastic approach for the nonlinear elastic response of the bio-tissue of interest. From single-step relaxation data of the urinary bladder, a fractional Maxwell model captures both short/long-term behaviors with two fractional orders, being the most suitable model for small strains at the first stage. For the second stage, multi-step relaxation data under large strains were employed to calibrate a four-parameter fractional quasi-linear viscoelastic model, that combines a Scott-Blair relaxation function and an exponential instantaneous stress response, to describe the elastin/collagen phases of bladder rheology. Our obtained results demonstrate that the employed fractional quasi-linear model, with a single fractional order in the range α = 0.25–0.30, is suitable for the porcine urinary bladder, producing errors below 2% without need for recalibration over subsequent applied strains. We conclude that fractional models are attractive tools to capture the bladder tissue behavior under small-to-large strains and multiple time scales, therefore being potential alternatives to describe multiple stages of bladder functionality.

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“…Here the fractional differential operator ∂ γ t := 0 I 1−γ t ∂ t for 0 ≤ γ < 1 with the fractional integral operator 0 I 1−γ t g := (t −γ /Γ(1 − γ)) * g and ∂ γ t := ∂ t u for γ = 1 [30]. Hence, the viscoelastic damping term −κ ∫ ∆Ω ∂ γ t udx accurately describes the behavior of viscoelastic damping, and includes the elastic resistance and viscoelastic damping as special cases [3,4,8,25,26,30,37,38], and has attracted growing research activities [9,11,13,17,18,21,24,28,31,32,41,42,44]. Consequently, the modeling equation becomes ∂ 2 t u(x, t) + κ∂ γ t u − K∇ 2 u(x, t) = f (x, t), (x, t) ∈ Ω × (0, T ], u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T ]; u(x, 0) = u 0 (x), ∂ t u(x, 0) =ǔ 0 (x), x ∈ Ω,…”

Section: Introductionmentioning

confidence: 99%

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

2021

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We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size $\sigma$ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom $N$ is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting $\sigma=N^{-1}$ leads to $O(N^3)$ operations of evaluating the temporal discretization coefficients and $O(N^2)$ memory. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only $O(\log N)$ terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of coefficients from $O(N^3)$ to $O(N^2\log N)$ and the corresponding memory from $O(N^2)$ to $O(N\log N)$. Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

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A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials

Cited by 32 publications

References 48 publications

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

A Data-Driven Memory-Dependent Modeling Framework for Anomalous Rheology: Application to Urinary Bladder Tissue

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

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