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Research Papers

A Reactive Inelasticity Theoretical Framework for Modeling Viscoelasticity, Plastic Deformation, and Damage in Fibrous Soft Tissue

[+] Author and Article Information
Babak N. Safa

Mem. ASME
Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716;
Department of Biomedical Engineering,
University of Delaware,
Newark, DE 19716
e-mail: safa@udel.edu

Michael H. Santare

Fellow ASME
Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716;
Department of Biomedical Engineering,
University of Delaware,
Newark, DE 19716
e-mail: santare@udel.edu

Dawn M. Elliott

Fellow ASME
Department of Biomedical Engineering,
University of Delaware,
Newark, DE 19716;
Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716
e-mail: delliott@udel.edu

1Corresponding author.

Manuscript received March 18, 2018; final manuscript received September 18, 2018; published online December 5, 2018. Assoc. Editor: Spencer P. Lake.

J Biomech Eng 141(2), 021005 (Dec 05, 2018) (12 pages) Paper No: BIO-18-1146; doi: 10.1115/1.4041575 History: Received March 18, 2018; Revised September 18, 2018

Fibrous soft tissues are biopolymeric materials that are made of extracellular proteins, such as different types of collagen and proteoglycans, and have a high water content. These tissues have nonlinear, anisotropic, and inelastic mechanical behaviors that are often categorized into viscoelastic behavior, plastic deformation, and damage. While tissue's elastic and viscoelastic mechanical properties have been measured for decades, there is no comprehensive theoretical framework for modeling inelastic behaviors of these tissues that is based on their structure. To model the three major inelastic mechanical behaviors of tissue's fibrous matrix, we formulated a structurally inspired continuum mechanics framework based on the energy of molecular bonds that break and reform in response to external loading (reactive bonds). In this framework, we employed the theory of internal state variables (ISV) and kinetics of molecular bonds. The number fraction of bonds, their reference deformation gradient, and damage parameter were used as state variables that allowed for consistent modeling of all three of the inelastic behaviors of tissue by using the same sets of constitutive relations. Several numerical examples are provided that address practical problems in tissue mechanics, including the difference between plastic deformation and damage. This model can be used to identify relationships between tissue's mechanical response to external loading and its biopolymeric structure.

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Figures

Grahic Jump Location
Fig. 1

Reference configuration of generations: in a body with the master reference configuration X for generation α the relative deformation gradient tensor (Fα) is defined using the reference configuration (Xα)

Grahic Jump Location
Fig. 2

Example of bonds with first-order rate equation: after a step deformation at t = 0, the bonds start to break (w0), and simultaneously reform to a new state (w1). With no further loading, all the bonds eventually break and reform to the new configuration as t→∞ (in this example, τ = 10).

Grahic Jump Location
Fig. 3

Consecutive step deformations and multiple generations: ((a)–(c)) number fraction of multiple generations initiated at four consecutive step deformations at t=0,15,30,45 with a first-order rate equation, and ((d)–(f)) their corresponding reference deformation gradients. ((a), (d) Formative bonds ((b), (e)) permanent bonds ((c), (f)) sliding bonds. For the number fraction graphs ((a)–(c)), the sum of all of the generations is shown with a horizontal green dashed line. The deformation stretch is shown with a red dashed line on reference configuration graphs. Note that Πα in general is a second-order tensor, and we used a one-dimensional representative for more convenient illustration ((d)–(f)).

Grahic Jump Location
Fig. 4

Effect of damage on kinetics: multiple generations initiated at four consecutive step deformations at t=0,15,30,45 with a first-order rate equation with damage shown for (a) formative bonds (b) permanent bonds and (c) sliding bonds, the sum of bonds declines after each step of loading until the final value of 0.75 (D = 0.25)

Grahic Jump Location
Fig. 5

Example 1: time response of formative bonds to a Heaviside step deformation in a large range of kinetics. The kinetics rate is controlled by time constant of reaction where (τf)0=10 and the range of time constants is covered by multiplying (τf)0 with powers of two. Formative bonds show an asymptotic decay in Cauchy stress during a sustained loading, whereas for large values of τf (slow kinetics rate) the response approaches a hyperelastic behavior (permanent bonds) marked with dashed line, and at small τf (high kinetics rate), the stress response approaches a singular behavior that is immediately decays to zero (sliding bonds). The step deformation is λ(t)=1.1u(t) and the neo-Hookean intrinsic hyperelasticity parameter is C1=100.

Grahic Jump Location
Fig. 6

Example 2: cyclic loading and damage (a) an increasing cyclic loading protocol applied for nondamaged (b) formative, (c) permanent, and (d) sliding bonds. When damage is added to the response for (e) formative, (f) permanent, and (g) sliding bonds, all the bond types show a softening behavior. The evolution of sliding and damage thresholds is also shown in (a) in response to loading and unloading phases, where sliding starts after λ=(r0)s=1.01, and damage after λ=(r0)D=1.03.

Grahic Jump Location
Fig. 7

Example 3: incremental stress relaxation with softening for ((a), (c)) formative and sliding bonds, and ((b), (d)) formative bonds with permanent bonds and damage. The effect of selection of either sliding bonds or permanent bonds is only observed during unloading. When the sliding bonds are used there is a shift in the reference configuration (c), which is not true for permanent bonds with damage (d).

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