For an application, the proposed nail growth model requires a choice of the mass-growth rate that (i) enables the nail matrix to increase in volume; (ii) considers that mass-growth is more dominant at the beginning of and within the proximal matrix while (iii) declines towards and beyond the end of the distal matrix; and (iv) underpins the postulate that mass-growth strain stays infinitesimally small at all times. These requirements are fulfilled by the choice
$$ r_{g} = g ( 1 - x_{1}/\hat{m} ) ( t_{0}/t ),\quad g \ll \rho_{0}/t_{0},\ \hat{m} > \ell, $$
(5.1)
which decreases longitudinally and considers that
\(g\) is a known, reasonably small positive constant.
5.1 Incompressible Mass-Growth of the Nail Matrix
In the case of incompressible mass-growth, where
\(\rho = \rho_{0}\) at all times, association of (
5.1) with (
3.6) yields
$$ u_{k,k} = \frac{gt_{0}}{\rho_{0}} ( 1 - x_{1}/\hat{m} )\ln (t/t_{0}), $$
(5.2)
while its combination with (
3.10) reveals that determination of the displacement components requires solution of the three uncoupled differential equations
$$ u_{1,jj} = \frac{ ( 1 + \lambda /\mu )gt_{0}}{\rho_{0}\hat{m}}\ln (t/t_{0}), \qquad u_{2,jj} = u_{3,jj} = 0. $$
(5.3)
The form (
5.2) of the volumetric strain reveals that solutions to (
5.3) may be sought in the form of simple, low degree polynomials of the co-ordinate parameters. Moreover, (
4.4b) and (
4.5) require from
\(u_{3}\) to be odd in
\(x_{3}\) and to cause no out of plane shear deformation. These requirements are met only if a trial solution of (
5.3c) has the form
\(u_{3} = cx_{3}\), where
\(c(t)\) is to be determined. However, the boundary condition (
4.6) returns
\(c = 0\) and, because this result leads to
the problem in hand reduces to one of plane strain. It follows that
\(u_{1}\) and
\(u_{2}\) are necessarily functions of
\(x_{1}\) and
\(x_{2}\) only and, as a result, indices in (
5.2) and (
5.3a, b) may take the values 1 and 2 only.
Inspection of the right hand sides of (
5.2) and (
5.3) suggests next that the simple polynomial solutions sought need not be more complicated than
$$ \begin{aligned}[c] &u_{1} = a_{1}x_{1}^{2} + b_{1}x_{2}^{2} + c_{1}x_{1}x_{2} + d_{1}x_{1} + e_{1}x_{2}, \\ &u_{2} = c_{2}x_{1}x_{2} + d_{2}x_{1} + e_{2}x_{2}, \end{aligned} $$
(5.5)
where the appearing coefficients may depend only on time. During the pre-critical stage of matrix growth, (
5.5a) should also satisfy the boundary condition (
4.7), which is replaced by the equation form of (
4.9) at
\(t = t^{\mathit{cr}}\).
5.2 The Pre-critical Stage of Matrix Mass-Growth: \(t< t^{\mathit{cr}}\)
Introduction of (
5.5) into (
5.2) and (
5.3a, b), as well as into the boundary condition (
4.7) and the corner condition (
4.8), yields the following intermediate results
$$\begin{aligned} &\begin{aligned}[c] &b_{1} = c_{1} = e_{1} = 0,\qquad a_{1} = \frac{ ( 1 + \lambda /\mu )gt_{0}}{2\rho_{0}\hat{m}}\ln (t/t_{0}),\qquad d_{1} = - a_{1}\ell, \\ &c_{2} = - 2a_{1} - \frac{gt_{0}}{\rho_{0}\hat{m}}\ln (t/t_{0}), \qquad e_{2} = - a_{1}\ell + \frac{gt_{0}}{\rho_{0}\hat{m}}\ln (t/t_{0}), \\ &d_{2} = - h_{\ell} ( c_{2} + e_{2}/\ell ), \end{aligned} \end{aligned}$$
(5.6)
use of which determines the non-zero displacement components as follows:
$$\begin{aligned} &\begin{aligned}[c] & u_{1} = \frac{ ( 1 + \lambda /\mu )gt_{0}}{2\rho_{0}\hat{m}}x_{1} ( x_{1} - \ell )\ln (t/t_{0}), \\ & \begin{aligned}[t] u_{2}& = \frac{gt_{0}}{2\rho_{0}\hat{m}} \biggl\{ - 2 \biggl( 2 + \frac{\lambda}{ \mu} \biggr)x_{1}x_{2} + \biggl( 3 - 2 \frac{\hat{m}}{\ell} + \frac{\lambda}{ \mu} \biggr)h_{\ell} x_{1} \\ &\quad {}+ \biggl( 1 + 2\frac{\hat{m}}{\ell} + \frac{\lambda}{\mu} \biggr)\ell x_{2} \biggr\} \ln (t/t_{0}). \end{aligned} \end{aligned} \end{aligned}$$
(5.7)
It is observed that, because
$$ u_{2}\vert _{x_{2} = 0} = \frac{gt_{0}}{2\rho_{0}\hat{m}} \biggl( 3 - 2\frac{\hat{m}}{\ell} + \frac{\lambda}{\mu} \biggr)h_{\ell} x_{1} \ln (t/t_{0}), $$
(5.8)
the nail matrix exerts its active mass-growth role by rising its bottom boundary if
\(\hat{m}/\ell < ( 3 + \lambda /\mu )/2\), or forcing it to fall if
\(\hat{m}/\ell > ( 3 + \lambda /\mu )/2\). That part of the specimen boundary remains thus intact through time only if
\(\hat{m}/\ell = ( 3 + \lambda /\mu )/2 = 3K/2 \mu\), where
\(K\) is the bulk modulus of the nail matrix material.
Some of the formulas that emerge in what follows obtain relatively simpler forms if the initial thickness of the nail matrix (Fig.
2) is assumed constant. In that case, in which
\(h_{0} ( x_{1} ) \equiv h_{\ell}\) in (
4.2b), the thickness of the growing, living part of the nail apparatus specimen is given as follows:
$$ h ( x_{1},t ) = h_{\ell} + u_{2}\vert _{x_{2} = h_{\ell}} = h_{\ell} \biggl\{ 1 + \frac{gt_{0}\ell}{2\rho_{0}\hat{m}} \biggl( 1 + 2\frac{\hat{m}}{\ell} + \frac{\lambda}{\mu} \biggr) ( 1 - x_{1}/\ell )\ln (t/t_{0}) \biggr\} . $$
(5.9)
In accordance with physical expectation, this is increasing with time, regardless of the direction that the bottom boundary of the matrix is forced to move. Thus the predicted, linearly decreasing longitudinal profile of the matrix thickness enables the corner condition (
4.8) and, subsequently, (
4.2a) to be satisfied at all times.
Combination of the displacement field (
5.7) and Hooke’s law (
3.8) yields next the non-zero stress components as follows:
$$ \begin{aligned}[c] &\sigma_{11} = \frac{gt_{0}\ell}{\rho_{0}\hat{m}} \biggl\{ \lambda \biggl( \frac{\hat{m}}{\ell} - 1 \biggr) + ( \lambda + 2\mu )\frac{x_{1}}{\ell} - \mu \biggr\} \ln (t/t_{0}), \\ &\sigma_{12} = \frac{gt_{0}}{\rho_{0}\hat{m}} \biggl\{ \biggl[ \lambda + \biggl( 3 - 2\frac{\hat{m}}{\ell} \biggr)\mu \biggr]\frac{h_{\ell}}{2} - ( \lambda + 2\mu )x_{2} \biggr\} \ln (t/t_{0}), \\ &\sigma_{22} = \frac{gt_{0}}{\rho_{0}} \biggl\{ ( \lambda + 2\mu ) + ( \lambda + \mu )\frac{\ell}{\hat{m}} - ( 3\lambda + 4\mu )\frac{x_{1}}{\hat{m}} \biggr\} \ln (t/t_{0}), \\ &\sigma_{33} = \frac{gt_{0}}{\rho_{0}} \biggl( 1 - \frac{x_{1}}{\hat{m}} \biggr)\ln (t/t_{0}). \end{aligned} $$
(5.10)
It may easily be verified that (
5.10) satisfy the equilibrium conditions (
2.2). This is evidently due to the fact that the expressions (
5.2) and (
5.3) obtained with the use of (
3.6) and (
3.10), respectively, ensure satisfaction of the Navier equations (
3.9).
It is thus concluded that completeness and uniqueness of the obtained linear elasticity solution require further that:
(a)
the part of the digit that lies beyond the left edge boundary of the nail specimen (Fig.
2), namely at
\(x_{1} \leq 0\), enables that boundary (
\(x_{1} = 0\)) to support the shear stress (
5.10b) and the normal stress
$$ \sigma_{11}\vert _{x_{1} = 0} = \frac{gt_{0}\ell}{ \rho_{0}\hat{m}} \biggl[ \lambda \biggl( \frac{\hat{m}}{\ell} - 1 \biggr) - \mu \biggr]\ln (t/t_{0}); $$
(5.11a)
(b)
the part of the proximal matrix that lies underneath the bottom boundary of the specimen (
\(x_{2} \leq\) 0) enables that boundary (
\(x_{2} = 0\)) to support the normal stress (
5.10c) and the shear stress
$$ \sigma_{12}\vert _{x_{2} = 0} = \frac{gh_{\ell} t_{0}}{2\rho_{0}\hat{m}} \biggl[ \lambda + \biggl( 3 - 2\frac{\hat{m}}{\ell} \biggr)\mu \biggr]\ln (t/t_{0}); $$
(5.11b)
(c)
the proximal nail fold (
\(x_{2} \geq h_{\ell}\)) enables the matrix top boundary (
\(x_{2} = h_{\ell}\)) to support the normal stress (
5.10c) and the shear stress
$$ \sigma_{12}\vert _{x_{2} = h_{\ell}} = - \frac{gh_{\ell} t_{0}}{2\rho_{0}\hat{m}} \biggl[ \lambda + \biggl( 1 + 2\frac{\hat{m}}{\ell} \biggr)\mu \biggr]\ln (t/t_{0}); $$
(5.11c)
(d)
combined action of the proximal nail fold (
\(x_{2} \geq h_{\ell}\)) and the part of the proximal matrix that lies underneath the bottom boundary of the specimen (
\(x_{2} \leq\) 0) balances the resultant shear traction caused by (
5.10b) on the matrix right boundary,
\(x_{1} = \ell\), where the boundary condition (
4.7) is already satisfied;
(e)
the part of the digit that lies beyond the lateral fold boundaries of the nail specimen
\((|x_{3}|\geq b)\) enables that boundary
\((|x_{3}|=b)\) to be free of shear traction and to support the normal stress (
5.10d).
If/when physically justified, potential modification of any of these boundary conditions, (a)–(e), will convert the mass-growth process considered in this section into a slightly or considerably different linear elasticity boundary value problem. This will need to be solved again and, hence, lead to displacement and stress distributions that necessarily differ from those detailed in (
5.7) and (
5.10), respectively. However, the principles that underpin the proposed nail elongation mechanism will still remain intact.
It is finally noted that, dependent on the value of the ratio
\(\hat{m}/\ell\),
\(\sigma_{11}\) may be negative and, therefore, compressive or positive and, therefore, tensile on the left edge of the matrix,
\(x_{1} = 0\) (Fig.
2). However,
\(\sigma_{11}\) is always positive in the neighbourhood of the tissue transitional plane (
\(x_{1} = \ell\)), where it reaches its maximum value
$$ \sigma_{11}\vert _{x_{1} = \ell} = \frac{gt_{0}}{\rho_{0}} ( \lambda + \mu \ell /\hat{m} )\ln (t/t_{0}) > 0. $$
(5.12)
This is thus tensile within the soft living part and, therefore, compressive within the hard dead part of the nail tissue. Moreover,
\(\sigma_{11}\) is increasing in time and, hence, is indeed set to reach, at some
\(t = t^{\mathit{cr}} > t_{0}\), a value that will trigger nail elongation through satisfaction of (
4.9) or, equivalently, (
4.10).
5.3 The Critical Stage: \(t= t^{\mathit{cr}}\)
Equation (
5.12) predicts that, at every instant of time,
\(\sigma_{11}\) is constant throughout the tissue transition plane
\(x_{1} = \ell\) and, hence, that either of (
4.9) or (
4.10) may conveniently obtain the simplified form
$$ \sigma_{11}\big\vert _{{\scriptstyle x_{1} = \ell \atop \scriptstyle t = t^{\mathit{cr}}}} = \sigma_{11}^{\mathit{cr}} = \frac{f_{1}^{\mathit{cr}}}{2bh_{\ell}}. $$
(5.13)
Connection of (
5.12) with (
5.13) yields then the time that the stress state reaches its critical level as
$$ t^{\mathit{cr}} = t_{0}\exp \biggl\{ \frac{\rho_{0}}{ ( \lambda + \mu \ell /\hat{m} )gt_{0}} \sigma_{11}^{\mathit{cr}} \biggr\} . $$
(5.14)
A subsequent substitution of this value back into (
5.10) provides the non-zero stresses developing within the nail matrix at
\(t^{\mathit{cr}}\) in terms of the known constant
\(\sigma_{11}^{\mathit{cr}}\). These are as follows:
$$ \begin{aligned}[c] &\sigma_{11}\vert _{t = t^{\mathit{cr}}} = \biggl\{ \lambda \biggl( \frac{\hat{m}}{\ell} - 1 \biggr) + ( \lambda + 2\mu ) \frac{x_{1}}{\ell} - \mu \biggr\} \frac{\sigma_{11}^{\mathit{cr}}}{ ( \mu + \lambda \hat{m}/\ell )}, \\ &\sigma_{12}\vert _{t = t^{\mathit{cr}}} = \biggl\{ \biggl[ \lambda + \biggl( 3 - 2\frac{\hat{m}}{\ell} \biggr)\mu \biggr]\frac{h_{\ell}}{2\hat{m}} - ( \lambda + 2\mu )\frac{x_{2}}{\hat{m}} \biggr\} \frac{\sigma_{11}^{\mathit{cr}}}{ ( \lambda + \mu \ell /\hat{m} )}, \\ &\sigma_{22}\vert _{t = t^{\mathit{cr}}} = \biggl\{ ( \lambda + 2 \mu ) + ( \lambda + \mu )\frac{\ell}{\hat{m}} - ( 3\lambda + 4\mu ) \frac{x_{1}}{\hat{m}} \biggr\} \frac{\sigma_{11}^{\mathit{cr}}}{ ( \lambda + \mu \ell /\hat{m} )}, \\ &\sigma_{33}\vert _{t = t^{\mathit{cr}}} = \biggl( 1 - \frac{x_{1}}{\hat{m}} \biggr)\frac{\sigma_{11}^{\mathit{cr}}}{ ( \lambda + \mu \ell /\hat{m} )}. \end{aligned} $$
(5.15)
In a similar context, (
5.7) reveals that the non-zero displacement components developing within the nail matrix when
\(t\) approaches
\(t^{\mathit{cr}}\), approach their limiting distributions
$$ \begin{aligned}[c] &\lim_{t \to t^{\mathit{cr}}}u_{1} = \frac{ ( 1 + \lambda /\mu )\sigma_{11}^{\mathit{cr}}}{2 ( \lambda \hat{m} + \mu \ell )}x_{1} ( x_{1} - \ell ), \\ & \begin{aligned}[t] \lim_{t \to t^{\mathit{cr}}}u_{2} &= \frac{\sigma_{11}^{\mathit{cr}}}{2 ( \lambda \hat{m} + \mu \ell )} \biggl\{ - 2 \biggl( 2 + \frac{\lambda}{\mu} \biggr)x_{1}x_{2} + \biggl( 3 - 2\frac{\hat{m}}{\ell} + \frac{\lambda}{\mu} \biggr)h_{\ell} x_{1} \\ &\quad {}+ \biggl( 1 + 2\frac{\hat{m}}{\ell} + \frac{\lambda}{ \mu} \biggr) \ell x_{2} \biggr\} . \end{aligned} \end{aligned} $$
(5.16)
However, those limiting displacement distributions will never be reached, because at
\(t^{c r}\) the nail matrix will bounce into the aforementioned alternative equilibrium state, described by some different displacement field,
\(\hat{\boldsymbol{u}}\).
The elastic energy that the nail matrix releases at
\(t = t^{\mathit{cr}}\) through that bouncing effect is
$$ U^{\mathit{cr}} = \frac{2b}{\mu} \int_{0}^{\ell} \int_{0}^{h(x_{1})} \biggl[ \sigma_{\alpha \beta} \sigma_{\alpha \beta} - \frac{\lambda}{3\lambda + 2\mu} ( \sigma_{\gamma \gamma} )^{2} \biggr]_{t = t^{\mathit{cr}}} dx_{2}dx_{1} = \bigl( \sigma_{11}^{\mathit{cr}} \bigr)^{2}2b\bar{U}^{\mathit{cr}} ( \lambda,\mu,h_{\ell},\ell,\hat{m} ), $$
(5.17)
where Greek indices take the values 1 and 2 only, and
\(h(x_{1})\) is evaluated by using (
5.9) at
\(t = t^{\mathit{cr}}\). It is noted that, because the critical stresses are proportional to
\(\sigma_{11}^{\mathit{cr}}\),
\(U^{\mathit{cr}}\) is proportional to
\((\sigma_{11}^{\mathit{cr}})^{2}\) and, hence,
\(\bar{U}^{\mathit{cr}} ( \lambda,\mu,h_{\ell},\ell,\hat{m} )\) is independent of
\(\sigma_{11}^{\mathit{cr}}\). With use of the critical stress distributions (
5.15), the form of
\(\bar{U}^{\mathit{cr}} ( \lambda,\mu,h_{\ell},\ell,\hat{m} )\) may be found analytically, or its value can be determined numerically for given values of the appearing arguments. The form or value of the function
\(\bar{U}^{\mathit{cr}} ( \lambda,\mu,h_{\ell},\ell,\hat{m} )\) is thus considered known in what follows.
5.4 Spring-Back Type of Nail Elongation
It is assumed that the anticipated spring-back action of the nail matrix takes place instantly at \(t^{c r}\); namely in a static manner, and without further interference of, or resistance from the hard nail plate. The nail matrix is thus temporarily perceived as an independent, strained and stressed elastic plate of length \(\ell\) which, by bouncing instantly into a new, unstrained and unstressed equilibrium state, increases its length and makes its thickness equal to that of its hard tissue counterpart; namely \(h_{\ell}\). The constant width of the plate, \(2b\), remains unchanged due to the prevailing plane strain conditions.
In a corresponding static equilibrium situation (
\(t = t_{0}\)), an initially unstressed and unstrained elastic rectangular plate that possesses the same geometric and material properties with those of the undeformed nail matrix can undergo an equivalent, mechanically caused spring-back jump if, after (i) its
\(x_{1} = 0\) edge is restrained against translation and (ii) its
\(x_{1} = \ell\) edge is subjected to a uniform compression,
\(- \sigma_{11}^{\mathit{cr}}\), that (iii) enables its stored energy to match the
\(U^{\mathit{cr}}\)-level given in (
5.17), (iv) it is allowed to bounce elastically back to its initial position through removal of the externally applied compression.
The relatively simple, exact elasticity solution to this reverse, linear elasticity plane strain problem is outlined in the
Appendix with use of standard Airy stress function techniques (e.g., [
18]). This shows that the bouncing type of displacements sought are averaged equivalents of
$$ \begin{aligned}[c] &u_{1}\vert _{t = t^{\mathit{cr}}} = \frac{\sigma_{11}^{\mathit{cr}}}{4\mu ( \lambda + \mu )} \bigl[ ( \lambda + 2\mu )x_{1} - \lambda h_{\ell} x_{2}/\ell \bigr], \\ &u_{2}\vert _{t = t^{\mathit{cr}}} = - \frac{\sigma_{11}^{\mathit{cr}}\lambda}{4\mu ( \lambda + \mu )} ( x_{2} - h_{\ell} x_{1}/\ell ), \end{aligned} $$
(5.18)
provided that their reverse counterparts (
A.6) store into the plate a total of strain energy that equals
\(U^{\mathit{cr}}\).
The elasticity solution obtained in the
Appendix reveals that the strain energy associated with the displacement field (
5.17) is
$$ U^{\mathit{cr}} = 2b\ell h_{\ell} \frac{ ( 1 + \lambda /\mu )}{2 + 3\lambda /\mu} \bigl( \sigma_{11}^{\mathit{cr}} \bigr)^{2}. $$
(5.19)
When compared with (
5.17), this yields
$$ \bar{U}^{\mathit{cr}} ( \lambda,\mu,h_{\ell},\ell,\hat{m} ) = 2\ell h_{\ell} \frac{ ( 1 + \lambda /\mu )}{2 + 3\lambda /\mu}, $$
(5.20)
which can be perceived as a non-linear algebraic equation for the essentially unknown parameter
\(\hat{m}\).
It is emphasised that the value of
\(\hat{m}\) obtained by solving (
5.20) depends on the elastic moduli and the geometric features of the living part of the nail tissue only. It is independent of, and, hence, irrelevant to the external frictions that caused
\(\sigma_{11}^{\mathit{cr}}\).
By releasing an amount of energy equal to that stored previously into the grown nail matrix, the displacement field (
5.18) brings the nail matrix into the unstressed and unstrained state described at the beginning of the
Appendix. Combination of (
5.18a) and (
4.11) yields thus the nail elongation as follows:
$$ \hat{L} = L + \frac{\sigma_{11}^{\mathit{cr}}}{4\mu ( \lambda + \mu )} \bigl[ ( \lambda + 2\mu )\ell - \lambda h_{\ell}^{2}/2\ell \bigr]. $$
(5.21)
It is noted in passing that an estimate of the thickness reached at
\(t = t^{\mathit{cr}}\) by the matrix part of the specimen depicted in Fig.
2 may similarly be obtained as follows:
$$ h^{\mathit{cr}} ( x_{1} ) = h_{\ell} + u_{2} \big\vert _{{\scriptstyle t = t^{\mathit{cr}}\atop\scriptstyle x_{2} = h_{\ell}}} = h_{\ell} \biggl\{ 1 + \frac{\sigma_{11}^{\mathit{cr}}\lambda}{4\mu ( \lambda + \mu )} ( 1 - x_{1}/\ell ) \biggr\} . $$
(5.22)
This approximate form of the critical thickness of the matrix may be used in this particular application for an estimate of the value of
\(\hat{m}\) in an alternative manner. Instead of employing the previously described strain energy matching, someone could accordingly attempt to match the critical thickness (
5.22) with the limiting expression obtained through (
5.9) when
\(t\) approaches
\(t^{\mathit{cr}}\).
Such a matching, which is here possible because both (
5.9) and (
5.22) vary linearly in
\(x_{1}\), yields
$$ \frac{\hat{m}}{\ell} = \frac{ ( \lambda + \mu )^{2} + \lambda \mu}{\lambda^{2} - 4\mu ( \lambda + \mu )}, $$
(5.23)
which, unlike its alternative estimate obtained by solving (
5.20), may return a value of
\(\hat{m}\) which does not depend on
\(h_{\ell}\). It is thus seen that, for positive values of
\(\lambda\) and
\(\mu\) that satisfy
$$ \lambda > 2\mu ( 1 + \sqrt{2} )\quad \Leftrightarrow \quad \nu > \frac{1 + \sqrt{2}}{3 + 2\sqrt{2}} \cong 0.414, $$
(5.24)
(
5.23) predicts that the ratio
\(\hat{m}/\ell\) is greater than 1 and, by virtue of (
5.1), it thus ensures that new mass is added within the nail matrix at all locations; here,
\(\nu\) represents Poisson’s the ratio of the proliferating material of the nail matrix.
However, the noted convenient matching of (
5.9) and (
5.22) will not be necessarily present in more complicated nail mass-growth applications, where derivation of a simple formula like (
5.23) may thus be not possible. In contrast, the energy release approach that leads to the algebraic Eq. (
5.20) is likely to work successfully always and, either analytically or numerically, to provide some admissible value for
\(\hat{m}\).