Chapter 4
Problems in Soft Tissue Biomechanics
Today, finite-element and other computational methods make it possible to solve virtually any problem in biomechanics and mechanobiology, no matter the complexity
of the problem. In many cases, however, much can be learned from a relatively
simple representation of the actual system. For example, early models for the left
ventricle consisted of spherical or cylindrical membranes or shells (Woods 1892;
Mirsky 1973; Yin 1981; Humphrey 2002). These models provided considerable
insight into fundamental behavior before more realistic models could be developed
(Wang et al. 2015). Often the results given by simple models are accurate enough for
many purposes, especially considering the variability inherent in biological systems.
This chapter considers some classical problems in nonlinear elasticity that have
proven useful in studies of soft tissues. They also can be used as benchmarks for
testing numerical codes. For simplicity, unless stated otherwise, body forces and
inertia are neglected.
For linear problems, it is common practice to prescribe the loads acting on
an object and compute deformation. In solving nonlinear problems analytically,
however, the reverse is often the case. Since constitutive relations define stress as
a nonlinear function of strain, it generally is easier to compute the loads required to
produce a prescribed deformation. This approach is called an inverse method, since
the result (deformation) is specified and the cause (loading) is computed. In a semiinverse method, only certain aspects of the deformation are specified a priori, with
the rest computed during the solution procedure.
In general, solving a problem in solid mechanics involves the following three
steps:
1. Analysis of kinematics (deformation, strain)
2. Analysis of stress (equations of motion, equilibrium)
3. Application of constitutive relations (stress-strain)
For the most part, we will follow these steps, although not necessarily in this order.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_4
155
Problems in Soft Tissue Biomechanics
Today, finite-element and other computational methods make it possible to solve virtually any problem in biomechanics and mechanobiology, no matter the complexity
of the problem. In many cases, however, much can be learned from a relatively
simple representation of the actual system. For example, early models for the left
ventricle consisted of spherical or cylindrical membranes or shells (Woods 1892;
Mirsky 1973; Yin 1981; Humphrey 2002). These models provided considerable
insight into fundamental behavior before more realistic models could be developed
(Wang et al. 2015). Often the results given by simple models are accurate enough for
many purposes, especially considering the variability inherent in biological systems.
This chapter considers some classical problems in nonlinear elasticity that have
proven useful in studies of soft tissues. They also can be used as benchmarks for
testing numerical codes. For simplicity, unless stated otherwise, body forces and
inertia are neglected.
For linear problems, it is common practice to prescribe the loads acting on
an object and compute deformation. In solving nonlinear problems analytically,
however, the reverse is often the case. Since constitutive relations define stress as
a nonlinear function of strain, it generally is easier to compute the loads required to
produce a prescribed deformation. This approach is called an inverse method, since
the result (deformation) is specified and the cause (loading) is computed. In a semiinverse method, only certain aspects of the deformation are specified a priori, with
the rest computed during the solution procedure.
In general, solving a problem in solid mechanics involves the following three
steps:
1. Analysis of kinematics (deformation, strain)
2. Analysis of stress (equations of motion, equilibrium)
3. Application of constitutive relations (stress-strain)
For the most part, we will follow these steps, although not necessarily in this order.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_4
155