Mathematical Modeling of Ligaments and Tendons

1993 ◽  
Vol 115 (4B) ◽  
pp. 468-473 ◽  
Author(s):  
S. L.-Y. Woo ◽  
G. A. Johnson ◽  
B. A. Smith
Keyword(s):  
Mathematical Modeling ◽  
Finite Strain ◽  
Mechanical Response ◽  
Time Dependent ◽  
Experimental Investigations ◽  
Display Time ◽  
Viscoelastic Models ◽  
The Past ◽  
Dependent Behavior ◽  
Single Integral

Ligaments and tendons serve a variety of important functions in maintaining the structure of the human body. Although abundant literature exists describing experimental investigations of these tissues, mathematical modeling of ligaments and tendons also contributes significantly to understanding their behavior. This paper presents a survey of developments in mathematical modeling of ligaments and tendons over the past 20 years. Mathematical descriptions of ligaments and tendons are identified as either elastic or viscoelastic, and are discussed in chronological order. Elastic models assume that ligaments and tendons do not display time dependent behavior and thus, they focus on describing the nonlinear aspects of their mechanical response. On the other hand, viscoelastic models incorporate time dependent effects into their mathematical description. In particular, two viscoelastic models are discussed in detail; quasi-linear viscoelasticity (QLV), which has been widely used in the past 20 years, and the recently proposed single integral finite strain (SIFS) model.

Viscoelastic Models for Ligaments and Tendons

1994 ◽  
Vol 47 (6S) ◽  
pp. S282-S286 ◽  
Author(s):  
S. L.-Y. Woo ◽  
G. A. Johnson ◽  
R. E. Levine ◽  
K. R. Rajagopal
Keyword(s):  
Constitutive Modeling ◽  
Finite Strain ◽  
Experimental Studies ◽  
Viscoelastic Behavior ◽  
Microstructural Change ◽  
Viscoelastic Models ◽  
Dependent Behavior ◽  
Recent Approach ◽  
Single Integral ◽  
Strain Model

Ligaments and tendons serve a variety of important functions in the human body. Many experimental studies have focused on understanding their mechanical behavior, mathematical modeling has also contributed important information. This paper presents a brief review of viscoelastic models that have been proposed to describe the nonlinear and time-dependent behavior of ligaments and tendons. Specific attention is devoted to quasi-linear viscoelasticity (QLV) and to our most recent approach, the single integral finite strain model (SIFS) which incorporates constitutive modeling of microstructural change. An example is given in which the SIFS model is used to describe the viscoelastic behavior of a human patellar tendon.

The Time Dependent Behavior of Linear Viscoelastic Polymers

2011 ◽  
Vol 675-677 ◽  
pp. 435-438
Author(s):  
Wei Xiang Zhang ◽  
Xing Shao ◽  
Zhao Ran Xiao
Keyword(s):  
Mechanical Behavior ◽  
Mechanical Characteristics ◽  
Mechanical Response ◽  
Constitutive Relations ◽  
Time Dependent ◽  
Numerical Examples ◽  
General Methodology ◽  
Inverse Transform ◽  
Dependent Behavior ◽  
Linear Viscoelastic

Polymers have been proved to have attractive mechanical characteristics, which made it desirable to choose these materials over traditional materials for numerous types of applications. As the uses of polymers increase, a thorough understanding of the mechanical behavior of these materials becomes vital in order to perform innovative and economical designs of various components. The main objective of this paper is to develop an effective method with the use of the Laplace inverse transform to describe the time dependent mechanical response of viscoelastic polymers. This general methodology is based on differential constitutive relations for viscoelastic polymers, avoiding the use of relaxation integral functions. As its application, the creep and relaxation properties of the materials are exhibited in the numerical examples.

Constitutive Modeling of Brain Tissue: Current Perspectives

2016 ◽  
Vol 68 (1) ◽  
Author(s):  
Rijk de Rooij ◽  
Ellen Kuhl
Keyword(s):  
Brain Tissue ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Constitutive Models ◽  
Time Dependent ◽  
Elastic Behavior ◽  
Integral Approach ◽  
Mechanical Stimuli ◽  
The Past ◽  
The Brain

Modeling the mechanical response of the brain has become increasingly important over the past decades. Although mechanical stimuli to the brain are small under physiological conditions, mechanics plays a significant role under pathological conditions including brain development, brain injury, and brain surgery. Well calibrated and validated constitutive models for brain tissue are essential to accurately simulate these phenomena. A variety of constitutive models have been proposed over the past three decades, but no general consensus on these models exists. Here, we provide a comprehensive and structured overview of state-of-the-art modeling of the brain tissue. We categorize the different features of existing models into time-independent, time-dependent, and history-dependent contributions. To model the time-independent, elastic behavior of the brain tissue, most existing models adopt a hyperelastic approach. To model the time-dependent response, most models either use a convolution integral approach or a multiplicative decomposition of the deformation gradient. We evaluate existing constitutive models by their physical motivation and their practical relevance. Our comparison suggests that the classical Ogden model is a well-suited phenomenological model to characterize the time-independent behavior of the brain tissue. However, no consensus exists for mechanistic, physics-based models, neither for the time-independent nor for the time-dependent response. We anticipate that this review will provide useful guidelines for selecting the appropriate constitutive model for a specific application and for refining, calibrating, and validating future models that will help us to better understand the mechanical behavior of the human brain.

scholarly journals Mathematical Modeling and Numerical Simulation of Atherosclerosis Based on a Novel Surgeon’s View

2021 ◽  
Author(s):  
Meisam Soleimani ◽  
Axel Haverich ◽  
Peter Wriggers
Keyword(s):  
Mathematical Modeling ◽  
Inflammatory Response ◽  
Phase Field ◽  
Mechanical Response ◽  
Type Equation ◽  
The Body ◽  
Vasa Vasorum ◽  
Diffusion Reaction ◽  
Dust Particles ◽  
Driving Mechanism

AbstractThis paper deals with the mathematical modeling of atherosclerosis based on a novel hypothesis proposed by a surgeon, Prof. Dr. Axel Haverich (Circulation 135(3):205–207, 2017). Atherosclerosis is referred as the thickening of the artery walls. Currently, there are two schools of thoughts for explaining the root of such phenomenon: thickening due to substance deposition and thickening as a result of inflammatory overgrowth. The hypothesis favored here is the second paradigm stating that the atherosclerosis is nothing else than the inflammatory response of of the wall tissues as a result of disruption in wall nourishment. It is known that a network of capillaries called vasa vasorum (VV) accounts for the nourishment of the wall in addition to the natural diffusion of nutrient from the blood passing through the lumen. Disruption of nutrient flow to the wall tissues may take place due to the occlusion of vasa vasorums with viruses, bacteria and very fine dust particles such as air pollutants referred to as PM 2.5. They can enter the body through the respiratory system at the first place and then reach the circulatory system. Hence in the new hypothesis, the root of atherosclerotic vessel is perceived as the malfunction of microvessels that nourish the vessel. A large number of clinical observation support this hypothesis. Recently and highly related to this work, and after the COVID-19 pandemic, one of the most prevalent disease in the lungs are attributed to the atherosclerotic pulmonary arteries, see Boyle and Haverich (Eur J Cardio Thorac Surg 58(6):1109–1110, 2020). In this work, a general framework is developed based on a multiphysics mathematical model to capture the wall deformation, nutrient availability and the inflammatory response. For the mechanical response an anisotropic constitutive relation is invoked in order to account for the presence of collagen fibers in the artery wall. A diffusion–reaction equation governs the transport of the nutrient within the wall. The inflammation (overgrowth) is described using a phase-field type equation with a double well potential which captures a sharp interface between two regions of the tissues, namely the healthy and the overgrowing part. The kinematics of the growth is treated by classical multiplicative decomposition of the gradient deformation. The inflammation is represented by means of a phase-field variable. A novel driving mechanism for the phase field is proposed for modeling the progression of the pathology. The model is 3D and fully based on the continuum description of the problem. The numerical implementation is carried out using FEM. Predictions of the model are compared with the clinical observations. The versatility and applicability of the model and the numerical tool allow.

Time-dependent behavior of nitric oxide excited states under the effect of electron impact

2021 ◽  
Vol 28 (2) ◽  
pp. 024503
Author(s):  
Mohammed amin Ferdi ◽  
Abdelaaziz Bouziane ◽  
Mourad Djebli
Keyword(s):  
Nitric Oxide ◽  
Electron Impact ◽  
Excited States ◽  
Time Dependent ◽  
Dependent Behavior
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