Behaviours of solid biological bodies are described in the context of solid mechanics. As is glanced in Chap. 1, load and deformation are characterized by stress and strain. They are introduced for one-dimensional mechanics in Sect. 1.2. First, this chapter gives their extension to three-dimensional mechanics under a small strain theory and a finite strain theory. It is followed by description of constitutive equations of linear elastic body and nonlinear hyperelastic bodies. Although biosolids behave as a viscoelastic body in general, the linear elasticity is a reasonably accounts for their behaviour as far as the deformation is small. Especially, the linear elastic theory works for hard biosolids such as bones in a physiological daily activity range. The linear elasticity is also useful for the first step of biomechanical analyses within the range of small deformation. For the finite deformations, a cyclic response in the physiological range is characterized by the concept of pseudo-elasticity with hyperelasticity. Second, constitutive equations are demonstrated for cortical bone and cancellous bone as a linear elastic body and for several soft tissues of an arterial wall, a skin and a cornea as typical examples of a nonlinear hyperelastic body. Third, an equilibrium problem is given as in the form of a virtual work and a stationary potential energy for linear elastic and nonlinear hyperelastic bodies. These provide us the basis for computational analyses of biosolids. Fourth, the fundamentals of a finite element method are given for a small strain theory and for a finite strain theory. Concept of the finite element approximation is explained for typical elements and the finite element equations are derived. Fifth, several computational biomechanics analyses are presented for orthopaedic, dental and ophthalmic biomechanics problems.
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Notes
- 1.
Cartesian coordinate system is employed throughout this chapter unless otherwise noted. That is, all the components are Cartesian components.
- 2.
Stress components \( {\sigma ^\prime_{{ij}}} \) in an \( {x^\prime_i} \)-coordinate system is calculated by using those \( {\sigma_{{ij}}} \) in an \( {x_i} \)-coordinate system as \( {\sigma ^\prime_{{ij}}} = \mathop{\sum }\limits_k \!\mathop{\sum }\limits_l {a_{{ik}}}{a_{{jl}}}{\sigma_{{kl}}} \), where \( {a_{{ij}}} \) is components of the rotation matrix, or \( ({\sigma ^\prime_{{ij}}}) = ({a_{{ik}}})({\sigma_{{kl}}}){({a_{{jl}}})^T} \)in matrix notation.
- 3.
Rules of summation convention are (1) each index can appear once or twice in any term and (2) every index appeared twice is summed over its range. For example, \( {a_{{ii}}} = \sum\limits_i {{a_{{ii}}}} = {a_{{11}}} +<$><$>{a_{{22}}} + {a_{{33}}} = \sum\limits_j {{a_{{jj}}} = {a_{{jj}}}} \), \( {a_{{ij}}}{b_j} = \sum\limits_j {{a_{{ij}}}{b_j}} = {a_{{i1}}}{b_1} + {a_{{i2}}}{b_2} + {a_{{i3}}}{b_3} \). Expression \( {a_{{ij}}}{b_i} = {c_j} \) and \( {a_{{ij}}}{b_j} + {c_{{ji}}}{d_j} \) are valid, but \( {a_{{jj}}}{b_j} \) and \( {a_{{ij}}}{b_j} = {c_j} \) are invalid.
- 4.
This is the Voigt notation in vector form.
- 5.
Kronecker-\( \delta \) tensor is defined as \( {\delta_{{ij}}} =\left\{ {\begin{array}{lll} 1 & {\hbox{when}} & {{{i}} = {{j}}} \\0 & {\hbox{when}} & {{{i}} \ne {{j}}} \\\end{array} } \right. \). It corresponds unit matrix, e.g. \( ({\delta_{{ij}}}) = {\left[ {\begin{array}{lllll} 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\\end{array}} \right] }\) in three dimensions, and replaces the index as \( {\delta_{{ij}}}{a_i} = {a_j} \).
- 6.
The ratio \( \frac{{{\rho_0}}}{\rho } \) is reciprocal of the volume ratio \( J = \frac{V}{{{V_0}}} \) of volume element deformed and undefromed because of conservation law \( {V_0}{\rho_0} = V\rho \). cf. (2.42).
- 7.
Here the strain energy density function \( W \) is defined first for a unit mass, then that for a unit volume is represented by \( {\rho_0}W \). The strain energy function for a unit volume comes first is an alternative as employed in other textbook.
- 8.
The stress–strain curve is strain-rate dependent and is approximated by Ramberg–Osgood model. That is, Young’s modulus is described as a power function of strain rate, but the effect of strain rate on it is as much as 15% in daily activity (Cowin 1989).
- 9.
The concept of a confidential interval in data fitted to the constitutive model and the problem of exterpolation are not limited to the constitutive models of cortical bone in this section. These points become more severely important for constitutive models for soft tissues exhibiting a strong nonlinearity found in the following sections.
- 10.
Throughout Sect. 2.3, notations in the original articles are used as much as possible in order to make further reference to the details easier.
- 11.
Green strain components \( {E_1} \) and \( {E_2} \) denotes the normal strain components \( {E_{{11}}} = {E_{{\theta \theta }}} \) in circumferential direction \( {x_1} \) and \( {E_{{22}}} = {E_{{zz}}} \) in longitudinal direction \( {x_2} \), respectively.
- 12.
This constitutive model uses the modified right Cauchy–Green tensor \( {\hat{C}_{{ij}}} \), and \( {\hat{{\rm I}}_l} \) denotes their invariants. Some fundamental variables in modified components are given by \( {F_{{ij}}} = {J^{{1/3}}}{\hat{F}_{{ij}}} \), \( {C_{{ij}}} = {J^{{2/3}}}{\hat{C}_{{ij}}} \), \( {\hat{C}_{{ij}}} = {\hat{F}_{{ki}}}{\hat{F}_{{kj}}} \), \( {E_{{ij}}} = {J^{{2/3}}}{E_{{ij}}} + \frac{1}{2}({J^{{2/3}}} - 1){\delta_{{ij}}} \) and \( {\hat{E}_{{ij}}} = \frac{1}{2}({\hat{C}_{{ij}}} - {\delta_{{ij}}}) \).
- 13.
The notation \( ({ }){,_i} \) stands for the partial derivative \( \frac{{\partial ({ })}}{{\partial {x_i}}} \) with respect to the coordinates \( {x_i} \). That is, \( {\sigma_{{ji}}}{,_j} = \frac{{\partial ({\sigma_{{ji}}})}}{{\partial {x_j}}} \) with summation convention.
- 14.
This is also called as a weak form.
- 15.
The body force \( {G_{{0i}}} \) per unit mass is explained at (2.36). The surface traction \( {P_{{0i}}} \) is such that \( {P_{{0i}}}{\hbox{d}}{S_0} = \frac{{\partial {x_j}}}{{\partial {X_m}}}{s_{{mi}}}{n_{{0i}}}{\hbox{d}}{S_0} \) for the boundary \( {\Gamma_{{0t}}} \) where \( {n_{{0i}}} \) is the surface unit normal to the undeformed configuration.
- 16.
In non-index notation for coordinate system, we have only the nodal index for \( u \) and \( x \). Displacement function is \( u(x) = {a_0} + {a_1}x \), and (2.132) is \( \left( {\begin{array}{lll} 1 & {{x_1}} \\1 & {{x_2}} \\\end{array} } \right)\left( {\begin{array}{lll} {{a_0}} \\{{a_1}} \\\end{array} } \right) = \left( {\begin{array}{lll} {{u_1}} \\{{u_2}} \\\end{array} } \right) \). Thus \( \left( {\begin{array}{lll} {{a_0}} \\{{a_1}} \\\end{array} } \right) = \frac{1}{{x{}_2 - {x_1}}}\left( {\begin{array}{lll} {{x_2}} & { - {x_1}} \\{ - 1} & 1 \\\end{array} } \right)\left( {\begin{array}{lll} {{u_1}} \\{{u_2}} \\\end{array} } \right) \) and shape functions are \( \Phi_1^{{(e)}} = \frac{{{x_2} - x}}{{{x_2} - {x_1}}} \) and \( \Phi_2^{{(e)}} = \frac{{x - {x_1}}}{{{x_2} - {x_1}}} \). These gives us the finite element approximation \( u(x) = \frac{{{x_2} - x}}{{{x_2} - {x_1}}}{u_1} + \frac{{x - {x_1}}}{{{x_2} - {x_1}}}{u_2} \) or \( u(x) = \left( {\begin{array}{lll} {\frac{{{x_2} - x}}{{{x_2} - {x_1}}}} & {\frac{{x - {x_1}}}{{{x_2} - {x_1}}}} \\\end{array} } \right)\left( {\begin{array}{lll} {{u_1}} \\{{u_2}} \\\end{array} } \right). \)
- 17.
We assume the existence of the inverse mapping \( {\xi_i} = {\xi_i}({x_j}) \).
- 18.
Summention convention is not applied for equations with ( )# in its equation number.
- 19.
Viscoelastic model of Kelvin-type is a parallel connection of a serial connection of a spring of constant \( {\mu_1} \) and a dashpot (damping component) of constant \( \eta \) and another spring of constant \( {\mu_2} \). The force-elongation relationship is written as \( F + {\tau_u}\dot{F} = {K_R}(u + {\tau_F}\dot{u}) \) where \( {K_R} = {\mu_2} \), \( {\tau_u} = {{\eta } \left/ {{{\mu_1}}} \right.} \) and \( {\tau_F} = ({{\eta } \left/ {{{\mu_2}}} \right.})[1 + {{{{\mu_2}}} \left/ {{{\mu_1}}} \right.}] \). The elongation \( c(t) \) for the force of a unit-step function \( 1(t) \) and the force \( k(t) \) for an elongation \( 1(t) \) are \( c(t) = {K_R}^{{ - 1}}[1 - (1 - {{{{\tau_u}}} \left/ {{{\tau_F}}} \right.})\exp ( - t/{\tau_F})]1(t) \) and \( k(t)=<$> <$>{K_R}[1 - (1 - {{{{\tau_F}}} \left/ {{{\tau_u}}} \right.})\exp( - t/{\tau_u})]1(t) \), respectively. These \( c(t) \) and \( k(t) \) are called the creep function and relaxation function.
- 20.
Kelvin-type continua is an extension of Kelvin-type viscoelastic model of springs and dashpot. Force and elongation are replaced with stress and strain as \( \sigma + {\tau_{\varepsilon }}\dot{\sigma } = {E_R}(\varepsilon + {\tau_{\sigma }}\dot{\varepsilon }) \) where \( {E_R} = {\mu_2} \), \( {\tau_{\varepsilon }} = {{\eta } \left/ {{{\mu_1}}} \right.} \) and \( {\tau_{\sigma }} = ({{\eta } \left/ {{{\mu_2}}} \right.})[1 + {{{{\mu_2}}} \left/ {{{\mu_1}}} \right.}] \). This is the model of one-dimensional Kelvin-type continuum, and stress–strain relationship is given by \( \sigma (t) = \int_0^t {k(t - \tau )} \dot{\varepsilon }(\tau ){\hbox{d}}\tau \) or \( \varepsilon (t) = \int_0^t {c(t - \tau )} \dot{\sigma }(\tau ){\hbox{d}}\tau \). This model is extended to three dimensions in the similar manner in one-dimensional elasticity to three-dimensional one, establishing the Kelvin-type continuum (Kelvin solid, also known as standard linear solid). It is noted here, for the Kelvin solid, the constants \( {\mu_1} \) and \( {\mu_2} \) are elastic moduli with the unit of Pa and \( \eta \) is viscosity with the unit of \( {\hbox{Pa}} \cdot {\hbox{s}} \).
- 21.
The original state of solid body in which no loads are applied is called the natural state or stress-free configuration. It is assumed zero stress and zero strain in the solid body. In the case of cornea, the in vivo state is a loaded state and the natural state will appear when the intraocular pressure is removed.
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Tanaka, M., Wada, S., Nakamura, M. (2012). Mechanics of Biosolids and Computational Analysis. In: Computational Biomechanics. A First Course in “In Silico Medicine”, vol 3. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54073-1_2
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