Analytical solutions to a network of standard linear solids

• Published in: Journal of engineering mathematics Vol. 105; no. 1; pp. 67 - 83
• Main Author: Carroll, Joseph E.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Biological materials; Computational Mathematics and Numerical Analysis; Constitutive equations; Constitutive relationships; Deformation; Differential equations; Eigenvalues; Formulas (mathematics); Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Mechanical properties; Real numbers; Strain; Theoretical and Applied Mechanics; Viscoelasticity; Viscoelasticity; Linear differential equation; Maxwell model; Kelvin–Voigt model; Maxwell–Wiechert model; Laplace transform; Standard linear solid
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-016-9882-6

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.