# Analytical solutions to a network of standard linear solids

Analytical solutions to a network of standard linear solids Various viscoelastic models, such as the standard linear solid, Maxwell model, and Kelvin–Voigt model, are frequently used to describe the behavior of biological materials from single cells to tissues. These models are expressed mathematically as simple differential equations, called constitutive equations, which relate the applied force (stress) to the resulting deformation (strain) of the material. Networks of these models, representing materials with heterogeneous mechanical properties, are described by systems of constitutive equations. We prove that the eigenvalues associated with such systems are all nonpositive real numbers, find bounds for them, and indicate how they can be estimated quickly and accurately. We then give formulas for the analytical solutions of the system of equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Engineering Mathematics Springer Journals

# Analytical solutions to a network of standard linear solids

, Volume 105 (1) – Feb 15, 2017
17 pages

/lp/springer_journal/analytical-solutions-to-a-network-of-standard-linear-solids-BAYhP1MXGK
Publisher
Springer Netherlands
Subject
Physics; Classical Mechanics; Applications of Mathematics; Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
0022-0833
eISSN
1573-2703
D.O.I.
10.1007/s10665-016-9882-6
Publisher site
See Article on Publisher Site

### Abstract

Various viscoelastic models, such as the standard linear solid, Maxwell model, and Kelvin–Voigt model, are frequently used to describe the behavior of biological materials from single cells to tissues. These models are expressed mathematically as simple differential equations, called constitutive equations, which relate the applied force (stress) to the resulting deformation (strain) of the material. Networks of these models, representing materials with heterogeneous mechanical properties, are described by systems of constitutive equations. We prove that the eigenvalues associated with such systems are all nonpositive real numbers, find bounds for them, and indicate how they can be estimated quickly and accurately. We then give formulas for the analytical solutions of the system of equations.

### Journal

Journal of Engineering MathematicsSpringer Journals

Published: Feb 15, 2017

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