# A predictive microstructure-based approach for the anisotropic damage,
residual stretches and hysteresis in biodegradable sutures

Gennaro Vitucci  DICATECh, Politecnico di Bari, Via Re David 200, 70125 Bari,
Italy  Domenico De Tommasi  DICATECh, Politecnico di Bari, Via Re David 200,
70125 Bari, Italy  Giuseppe Puglisi  DICATECh, Politecnico di Bari, Via Re
David 200, 70125 Bari, Italy  Francesco Trentadue 111Corresponding author:
<EMAIL_ADDRESS>DICATECh, Politecnico di Bari, Via Re David
200, 70125 Bari, Italy

###### Abstract

We propose a predictive model for the mechanical behavior of biodegradable
polymers of interest for biomedical applications. Starting from a detailed
description of the network behavior of the copolymer material, taking care of
bonds breaking and recrosslinking effects, folded-unfolded transitions and
network topological constraints, we deduce a macroscopic law for the complex
mechanical behavior of many biomedical materials with a particular focus on
absorbable suture threads and a perspective to new material design. A crucial
novelty of the model is the careful description of the observed microscopic
anisotropic damage induced by the deformation, here described based on the
classical microsphere integration approach. The resulting energy,
characterized by a few number of material parameters, with physically clear
interpretation, is successfully adopted to predict our cyclic experiments on
poliglecaprone sutures with anisotropic damage, permanent stretch and internal
hysteresis. By also studying other suture materials behavior we show that the
predictivity properties of the model can be extended also to materials with a
different mechanical response and thus also possibly applied for the design of
new, high-performance biomedical materials.

Keywords— Sutures threads, Mullins effect, Biodegradable copolymers,
Micromechanically-based model, Worm-like chain, Damage induced anisotropy

##  1 Introduction

Biodegradable copolymers are successfully adopted in different biomedical
applications. In particular synthetic polymers such as polylactic acid (PLA),
polyglycolic acid (PGA), and polycaprolactone (PCL) have been adopted for
their biocompatibility and biodegradability. With specific reference to suture
applications, beyond the quoted properties, also the mechanical response of
the material under cyclic loading is fundamental in the everyday medical
practice. The prediction of residual stretches, toughness, stress softening,
good tensile and knot-pull strengths are crucial in clinical applications. The
effective microstructure characterization of these materials is then key both
for predicting the behavior of existing suture threads and for the design of
new engineering materials with prescribed macroscopic material response.

In this paper, we extend our previous approach proposed in [1] by accounting
for a fully three-dimensional point-wise distribution of polymeric chains
whose properties vary with stretch history. As well known, soft materials are
characterized by a fundamental anisotropy of damage since each chain natural
configuration depends on its own extension history. Specifically, in the
framework of statistical mechanics, we propose a constitutive approach, in
which the Helmholtz free energy is composed of two terms: an entropic Worm
Like Chain (WLC) component and a network topological one. An important novelty
of the WLC term is that, to consider the important effect of unloading, we
assume, again based on statistical mechanics results, that both the natural
and contour length of each chain depend on molecule stretch histories. In
particular we describe the copolymer molecules as a two phase material with
hard (folded) and soft (amorphous) domains. Due to externally imposed
stretching, folded domains undergo a hard$\rightarrow$soft transition with an
increase of available monomers producing a contemporary growth of the contour
and natural lengths. These processes typically involve exchange of heat with
the environment due to formation and dissociation bond enthalpy and as such
should rigorously be viewed as complex multiphysics phenomena [2, 3]. For the
sake of simplicity and in order to deal only with purely mechanical
quantities, we instead describe bonds rupture and recrosslinking and the
resulting microscopic dissipation in an isothermal setting. By following the
approach proposed in [4, 5] we then introduce a second important energy term,
taking care of the network topological constraint. Based on the deduced
filament Helmholtz free energy, we consider, by following the classical
microsphere approach [6, 5], an isotropic chains distribution of the
copolymer. As a result, for an assigned deformation history, we are able to
deduce the chain stretch in each direction that, after numerical integration,
gives back the total internal specific energy with history dependent
parameters, describing the anisotropic damage effect. The resulting
macroscopic model is characterized by a set of five parameters only: one for
each component of the free energy, one relating the contour length to the
natural end-to-end chain length and two describing refolding and unfolding in
unloading and reloading. In this way we are able to capture both anisotropic
damage, residual stretches, and hysteresis.

As a consequence, a micromechanically-based model for biodegradable isotropic
polymers is developed and validated with experimental data on biodegradable
copolymer suture threads. The comparison with our experiments on
poliglecaprone 25 (Monocryl® [7]) shows the excellent capacity of the model in
predicting the experimental behavior. To further support the proposed model,
in A we successfully reproduce the cyclic behavior of different absorbable
sutures by introducing a further parameter for the internal hysteresis. The
resulting model exhibits significant improvements, as compared with previous
models in the literature [8, 9] in the prediction of the sutures experimental
behavior under cyclic loading.

##  2 Methods

Figure 1: Cartoon of the chain deformation $\lambda$ decomposition into an
elastic and a natural stretch $\lambda_{e}$ and $\lambda_{n}$, respectively.
The light blue tube represents the topological constraint arising from the
chain interaction within the network. The amorphous region length (thin red
line outside the tube) does not contribute to the filament length $L$ since it
is expected to be small compared to the unfolded region (hidden length).
Figure 2: Filament loading history. Inset) Stretch vs time. The magenta line
shows the primary path, the path AB unloading, the path BC reloading (see
Eq.8). Main) Force vs stretch. An internal hysteresis cycle (gray) is produced
along the path ABC. The residual stretch $\bar{\lambda}_{n}$ is the stretch at
zero force and evolves with the primary path.

In this section we extend a model for biodegradable sutures proposed by some
authors of this paper [1] in several directions. On one side we consider the
important anisotropic effect in the evolution of the natural molecules
configurations and the topological network constrains, on the other side, by
introducing recrosslinking effect, we describe the important experimental
effect of internal hysteresis cycles. Similarly we describe the insurgence of
residual stretches fundamental for modelling the clinical application of the
suture.

Our model starts from the observation of the behavior at the molecular scale.
While our previous model was based on a simplifying assumption of non-
interacting molecules in the direction of the suture, here we consider a more
detailed analysis considering of a three-dimensional network of chains
undergoing cyclic uniaxial stress histories.

As in usually adopted schemes of rubber elasticity, our multiscale model is
based on the affinity assumption identifying macroscale stretches with the
molecular ones [10]. In particular our approach is based on the microsphere
integration scheme [6], firstly applied by [5, 11, 12] to model polymer
networks mechanical behavior. In this way, as we show in the following, we are
able to describe in detail the evolution of anisotropic damage and residual
stretches.

Our microstucture scheme is represented in Fig.3. As described in [13, 14, 15,
16] copolymers molecules are constituted by alternating sequences of coiled
and folded domains, here interpreted as two different material phases. We then
deduce the macroscale material behavior as an homogenized effect of complex
links breaking and recrosslinking [17, 18]. Roughly speaking, we can imagine
that as we apply a given deformation the hard domains of elongated chains
undergo unfolding, corresponding from one side to a hard$\rightarrow$soft
material transition, from the other side to a variation of the molecule
expected length, due to the availability of a larger number of free monomers.
When unloaded the chains refold to simple entropic effects and consequently
some broken links can rebind.

As described for instance in [19, 20, 5, 21, 22] these effects are strongly
anisotropic, so that an effective description of the material behavior
requires accounting for different elongations in different directions and this
is at the base of the proposed predictive model for suture materials.

###  2.1 Single chain

Following the approach in [1], we base our model on Flory’s assumption that
the expected end-to-end length $L_{n}$ of the co-polymer macromolecules
composed of $n$ Kuhn segments of length $b$ can be estimated, based on
classical statistical mechanics (see [23]), as

$L_{n}^{2}=C_{\infty}nb^{2},$ (1)

where Flory’s characteristic ratio $C_{\infty}$ is unitary for ideal Gaussian
chains, whereas it grows with the persistence length [10].

To apply the recalled affinity assumption we may introduce the following
stretch of the macromolecules:

$\lambda=\frac{L}{L_{0}},\quad\lambda_{n}=\frac{L_{n}}{L_{0}},\quad\lambda_{c}=\frac{L_{c}}{L_{0}}\quad$
(2)

where $L_{0}$ is the expected length of the previously unloaded chain, whereas
$L_{c}=nb$ is the contour length, measuring the limit chain extensibility.
Interestingly, by using Eq.1, we may observe [1] that the contour stretch
$\lambda_{c}$ is related to the natural one $\lambda_{n}$ via the initial
contour stretches $\lambda_{co}$:

$\lambda_{c}=\lambda_{co}\lambda^{2}_{n},\qquad\lambda_{co}=\frac{L_{0}}{C_{\infty}b}.$
(3)

Observe that, in accordance with classical finite elasto-plasticity, the total
stretch $\lambda$ is multiplicatively decomposed (see Fig.1) into an elastic
and a permanent part

$\lambda=\lambda_{e}\lambda_{n}.$ (4)

As a result, the model has a single internal variable describing inelastic
behavior that we choose as the natural stretch $\lambda_{n}$. Notice that we
neglect the length of the folded domains thus assuming the existence of the
so-called hidden length that is involved in the model due to the rupture of
sacrificial bonds during unraveling as commonly understood in biopolymers
mechanics [24].

In this work we consider for the chain the following Helmholtz free energy
density (per unit contour length)

$\psi(\lambda,\lambda_{n})=\kappa\dfrac{\left(\lambda-\lambda_{n}\right)^{2}}{\lambda_{c}-\lambda}-\nu\lambda\left(1+\lambda_{e}^{-2}\right),$
(5)

where the first term describes the worm-like chain energy type used in [1]
whereas the second one has been proposed in [13, 25] and captures a
topological constraint arising from the chain entanglement with its
environment and itself. This term is expected to be small for semiflexible
chains. Loosely speaking, this term takes care of a Poisson type confinement
effect for a chain elongating inside the surrounding network.The
proportionality with $\lambda_{e}^{-2}$ is linked to the aerial deformation.
Indeed, at given $\lambda_{n}$ this energy terms decreases as $\lambda_{e}$
grows to describe a decreasing entropic term that may be loosely interpreted
as a lower energy required to elongate chains constrained in thinner tubes.

The first principle of thermodynamics for our isothermal system gives

$f\dot{\lambda}(t)=\dot{\psi}(\lambda_{n}(t),\lambda(t))+\gamma(t),$ (6)

where $\gamma$ is the rate of energy dissipation and $t$ is a time type
variable describing the loading history in the proposed rate-independent
behavior. Following the classical settings of [26] for thermodynamical systems
with internal state variables, we find the equilibrium force $f$ as

$\begin{split}f&=\frac{\partial\psi}{\partial\lambda}=f_{\kappa}+f_{\nu}\\\
&=\kappa\left[\left(\frac{\lambda_{c}-\lambda_{n}}{\lambda_{c}-\lambda}\right)^{2}-1\right]-\nu\left[1-\lambda_{e}^{-2}\right].\end{split}$
(7)

As anticipated, here we consider the possibility that, as in the case of knots
formation and/or other clinical conditions, the suture can undergo cyclic
force loading as represented in Fig.2. Next, we assume the following
phenomenological evolution of $\lambda_{n}$ depending on loading history,
which can be distinguished into the following conditions (see Fig.1):

$\left\\{\begin{array}[]{llll}\lambda=\lambda_{\text{max}},&\dot{\lambda}>0&...&\text{primary
path}\\\ \lambda=\lambda_{\text{min}},&\dot{\lambda}<0&...&\text{unloading
path}\\\
\lambda_{\text{min}}<\lambda<\lambda_{\text{max}},&\dot{\lambda}>0&...&\text{reloading
path}\\\ \end{array}\right.$ (8)

To attain an explicit analytic description of the complex breaking and
recrosslinking phenomena, we consider special memory properties of the
material. Specifically, we assume that on the primary path the macromolecule
unfolds depending on the maximum attained stretch $\lambda_{max}$. On the
other hand, as anticipated, we assume that under unloading there is a partial
rebinding and under reloading a new unbinding of reformed links. Furthermore
the memory of the minimum stretch $\lambda_{\text{min}}$ is reset at every
cycle.

The evolution of breaking and recrosslinking effects during loading, unloading
and reloading depends on the specific properties of the chains and of the
network. Here, based on an energetic interpretation proposed in [13], we
assume the following evolution law for the internal variable

$\lambda_{n}=\sqrt{\lambda_{\text{max}}}\left(\frac{\lambda}{\lambda_{\text{max}}}\right)^{{\beta_{u}+\beta_{r}\sqrt{\lambda-\lambda_{\text{min}}}}}\geq\bar{\lambda}_{n}\quad$
(9)

depending on two only positive scalar material parameters $\beta_{u}$ and
$\beta_{r}$ measuring the unfolding and refolding evolution.

In particular, during primary loading we get

$\lambda_{n}=\sqrt{\lambda_{\text{max}}},\quad\lambda_{c}=\lambda_{co}\lambda_{max}$
(10)

that can be obtained using Eq.3 under the assumption that $L_{c}\propto
L_{max}$ descending by the hypothesis of a constant dissipation rate per unit
elongation. Notice that in view of Eq.8 and 9 we have a purely reversible
elastic response in compressed fibers.

The second factor in Eq.9 accounts for internal hysteresis effects.
Specifically, the zero-force residual stretch $\bar{\lambda}_{n}$ of Eq.9 is
determined by imposing the condition $\lambda=\lambda_{n}$, which gives

$\bar{\lambda}_{n}=\lambda_{\text{max}}{}^{\frac{1}{2}\frac{1-2\beta_{u}}{1-\beta_{u}}},$
(11)

i.e. the residual stretch depends on the maximum historic stretch in the
particularly simple form of a power law. This prospects the possibility of
experimentally determining the exponential factor $\beta_{u}$ and moreover
assigns also the variation of the contour stretch
$\bar{\lambda}_{c}=\lambda_{co}\bar{\lambda}_{n}^{2}$.

To continue with a microstructure interpretation of Eq.9, during an internal
cycle ${\lambda}_{min}(=\bar{\lambda}_{n})\leq\lambda<\lambda_{max}$ as in
Fig.2, there exists a fraction $\rho$ of reversibly unfolded Kuhn segments
with respect to the cumulative number of unfolded monomers. Specifically,
since the contour stretch is proportional to the total number of unfolded
segments $n$, via Eq.3, Eq.11, and Eq.10 we obtain

$\rho(\lambda_{\text{max}})=\frac{\hat{\lambda}_{c}(\lambda_{\text{max}})-\bar{\lambda}_{c}}{\hat{\lambda}_{c}(\lambda_{\text{max}})}=1-\lambda_{\text{max}}{}^{-\frac{\beta_{u}}{1-\beta_{u}}},$
(12)

where the function $\lambda_{c}=\hat{\lambda}_{c}(\lambda_{\text{max}})$ is
determined by Eq.3 and Eq.9. Observe that the recovering increases with
$\lambda_{max}$ and this will be confirmed by the following experimental
analysis. Moreover, after reloading, when the stretch reaches again its
previously attained maximum value $\lambda=\lambda_{\text{max}}$, these
recovered fraction is again unfolded. In this way the internal hysteresis
equals the unfolding energy of this fraction. Eventually, the system memory
respects the Return Point Memory property [27], because when
$\lambda=\lambda_{\text{max}}$ the system goes back exactly to the starting
configuration in the primary loading path.

Thus, the two additional parameters $\beta_{u}$ and $\beta_{r}$ describe
internal hysteresis. In particular $\beta_{u}$ measures refolding properties
during unloading, thus influencing also the unfolding variation of natural
length, whereas $\beta_{r}$ measures the hysteresis size due to refolding
during reloading ([13, 14, 15]). All things considered, our single chain model
is able to describe the damage, the hysteresis and the residual stretches in
loading unloading and reloading via only five parameters.

#### Thermodynamic consistency

We recall that our modeling is inscribed in the framework of thermodynamics
with internal state variables in the special simple setting of isothermal
processes [26], so that we do not consider the contribute of the heat flux to
the dissipation rate. From Eq.6 and Eq.7 one obtains that the rate of
dissipation $\gamma(t)$ of a single fiber in the particular case of an
isothermal process is

$\gamma(t)=-\frac{\partial\psi}{\partial\lambda_{n}}\dot{\lambda}_{n}\geq
0\;.$ (13)

In view of Eq.5 and Eq.3 it reads

$\begin{split}\gamma(t)=\left(2\kappa\dfrac{\lambda_{e}(\lambda-\lambda_{n})}{\lambda_{c}-\lambda}+2\dfrac{\nu}{\lambda_{e}}\right)\dot{\lambda}_{n}\geq
0\end{split}\;.$ (14)

For fibers in compression $(\lambda<\lambda_{r})$ we have considered a purely
elastic response, then $\dot{\lambda}_{n}=0$ and Eq.14 is clearly satisfied.
For fibers in traction $(\lambda>\lambda_{r})$ we always have

$\lambda_{n}\leq\lambda<\lambda_{c},$ (15)

and then (13) is satisfied if the elastic constants $\kappa$ and $\nu$ are
positive and $\dot{\lambda}_{n}>0$, whereas is not satisfied in an unloading
path. However, as typical for hysteresis cycles, the global condition

$\displaystyle\oint_{\text{cycle}}\\!{f(\lambda)\dot{\lambda}}\,\mathrm{d}t\geq
0,$ (16)

granting dissipative internal cycles, is numerically verified and regulated by
the parameter $\beta_{r}$. We may interpret this result as due to the physical
condition that part of the heat production accompanying links scissions can be
reused by the system to partial healing and refolding, with a global
thermodynamic consistency of each cycle.

###  2.2 Network model

Figure 3: Micro-macro scale transition. The representative volume (light blue)
contains a distribution of filaments which are isotropically oriented in space
along the unit vector $\mathbf{D}$ whose directional cosines are measured on
the microsphere (yellow). The macroscopic deformation gradient $\mathbf{F}$
stretches and rotates the filaments according to the affinity hypothesis.

To model the macroscopic thread behavior, we assume that it is composed by a
network of isotropically oriented chains in the framework of rubber elasticity
[5]. Furthermore, by the classical affinity assumption, we correlate the chain
deformation with the macroscopic one. Thus, if $\mathbf{F}$ is the macroscopic
deformation gradient, a (unitary length) chain oriented along $\mathbf{D}$ in
the reference configuration is transformed in the present configuration into
the deformed vector

$\mathbf{d}=\mathbf{F}\mathbf{D},\qquad\lambda=\|\mathbf{d}\|.$ (17)

Under the considered isotropic assumption, the probability that a chain points
in the direction associated to $\mathbf{D}$ is $1/4\pi$ and then the mean
total free energy of the microstructure is

$\Psi=\frac{1}{4\pi}\int_{\text{sphere}}\\!{\psi(\lambda,\lambda_{n})}\,\mathrm{d}\Omega\;.$
(18)

As in [26], one can obtain the first Piola-Kirchhoff stress tensor
$\mathbf{\Sigma}$ as

$\mathbf{\Sigma}=\frac{\partial\Psi}{\partial\mathbf{F}}.$ (19)

#### Macroscopic uniaxial extension

While our model can be applied to generic deformation classes, such as in the
case of biological tissues, in this paper we focus on the case of uniaxial
extension in the thread direction, because this is the case of interest in
biomedical applications.

Thus, under the classical hypothesis of incompressible materials, i.e.
$\det\mathbf{F}=1$, the deformation gradient in a principal reference frame
takes the form

$\mathbf{F}=\left[\begin{matrix}\Lambda&0&0\\\ 0&\Lambda^{-\frac{1}{2}}&0\\\
0&0&\Lambda^{-\frac{1}{2}}\\\ \end{matrix}\right].$ (20)

Since the work on the system is performed by the only non-zero component
$\sigma$ of $\mathbf{\Sigma}$, i.e. the normal traction, similarly to Eq.19 it
results

$\sigma=\frac{\partial\Psi}{\partial\Lambda}=\frac{1}{4\pi}\int_{\text{sphere}}\\!{\frac{\partial\psi}{\partial\lambda}\frac{\mathrm{d}\lambda}{\mathrm{d}\Lambda}}\,\mathrm{d}\Omega.$
(21)

Integrals on the microsphere of the type (18) can be computed via the
numerical scheme firstly proposed by [6]

$\begin{split}\langle
v\rangle=\frac{1}{4\pi}\int_{\text{sphere}}\\!{v(\mathbf{D})}\,\mathrm{d}\Omega\approx\sum_{i=1}^{N}w_{i}v_{i},\\\
\quad v_{i}=v(\mathbf{D}_{i}),\quad\sum_{i=1}^{N}w_{i}=1,\end{split}$ (22)

where $w_{i}$ are weights and $N$ is the number of directions in which the
sphere is numerically decomposed (in this work we use a number of directions
$N=33$). Thus we have

$\sigma\approx\sum_{i=1}^{N}w_{i}\frac{\mathrm{d}\psi_{i}}{\mathrm{d}\Lambda}=\sum_{i=1}^{N}w_{i}f(\lambda_{i})\frac{\mathrm{d}\lambda_{D_{i}}}{\mathrm{d}\Lambda},$
(23)

where the function $f(\lambda_{i})$ has been given in Eq.7.

Note that in our isochoric case we directly derive $\sigma$ from a mechanical
energy balance where pressure does not appear. This procedure is
computationally cheaper than the general one (see [5, 11, 12]), where
integration of microstresses is needed in order to compute all the nonzero
components of the macrostress.

##  3 Model validation and discussion

Observe that very different mechanical behaviors can be attained regarding
hardening, permanent stretches, and hysteresis. All these properties can be
crucial in biomedical applications and since our parameters have a clear
microstructure interpretation we believe that our model can also help in the
direction of designing new bioinspired materials.

In the following, to validate the proposed model in capturing the experimental
behavior of sutures, with complex cyclic loading and taking care of damage
(Mullins effect) and residual stretches, together with internal hysteresis we
refer to our experiments, previously described in [1] by the simpler model
recalled in the introduction, on diffuse synthetic absorbable Monocryl®
(poliglecaprone 25) monofilament sutures threads (see Fig.4a).

The vector of unknown constitutive parameters
$\mathbf{v}=(\kappa,\mu,\lambda_{c0},\beta_{u},\beta_{r})$ is determined by
minimizing the error function:

$\mbox{Err}(\mathbf{v})=\dfrac{|\bm{\sigma}_{exp}-\bm{\sigma}(\mathbf{v})|\,}{|\bm{\sigma}_{exp}|}\,.$
(24)

Here $\left|\cdot\right|$ is the 1-norm, $\bm{\sigma}_{exp}$ is the extracted
sample of experimental values of stress and $\bm{\sigma}(\mathbf{v})$ is the
vector of corresponding analytic values computed as in Eq.23. Next, the
following optimization problem is posed

$\begin{cases}\min\mbox{Err}(\mathbf{v})\\\
\mathbf{v}^{l}\leq\mathbf{v}\leq\mathbf{v}^{u}\\\ \end{cases}$ (25)

where $\mathbf{v}^{l}$ and $\mathbf{v}^{u}$ are lower and upper bounds for the
constitutive parameters, derived from physical considerations. A global
optimization procedure has been adopted, using the particleswarm toolbox of
MATLAB® and adopting a population size of 400 elements. The obtained optimal
values of the constitutive parameters are reported in Tab.1.

$\kappa$[$10^{8}\times$Pa] |  $\nu$[$10^{8}\times$Pa] | $\lambda_{c0}$ | $\beta_{u}$ | $\beta_{r}$
---|---|---|---|---
0.148 | 3.548 | 1.008 | 0.825 | 0.359
Table 1: Parameter identification for Monocryl. Error of the optimization
procedure 2.62$\%$.

In Fig.4a a comparison between our model and the experimental data is shown.
The error determined by Eq.24 is $\mbox{Err}(\mathbf{v}_{opt})=2.62\%$, mainly
regarding an experimental non perfect respect of the Return Point Memory
assumption. Interestingly, despite the complication due to the introduction of
internal hysteresis, we improved our previous results [1] at fixed number of
parameters thanks to both the adoption of the three-dimensional distribution
of chains and the introduction of the topological term.

Figure 4: Comparison of the experimental and theoretical stress-strain
diagrams. Our experimental data, model parameters as in Table 1. a) Overall
performance of the current model. b) Separate contributions of WLC and
topological stresses. c) Same as b) plotted as a function of the time order
parameter $t$ highlighting that the topological stress gives maximum relative
contribution toward low stresses.

Specifically, in Fig.4b we decomposed the contribution of the two additive
parts of the WLC chains $\sigma_{\kappa}$ and of the topological constraint
$\sigma_{\nu}$, where $t$ is a pseudo time parameter. It can be noted that,
even if the elastic constant $\nu$ is much greater than $\kappa$,
$\sigma_{\nu}$ is much smaller than $\sigma_{\kappa}$ and is always negative
in accordance with our previously introduced physical interpretation. In
particular the analysis of the numerical results suggests that this term plays
a more important role in the description of the unloading states of the
material. Moreover, it is evident from this numerical analysis that the
internal hysteresis dissipation quantified as in Eq.16 is produced chiefly by
the WLC part of the stress, as expected from the fact that the topological one
only contains the elastic stretch $\lambda_{e}$ (see Fig.4.c, green curve).

Figure 5: Damage induced anisotropy. Microstructure uniaxially loaded along
the north-south direction. Points on the microsphere and related projection
onto the horizontal plane (blue, middle panel) show the $2\times 33$ direction
cosines used for integration. a) Maximum damage per latitude on the
microsphere. The filaments whose end-to-end unit vector is tilted less than
$\approx 30^{\circ}$ with respect to the horizontal result undamaged, since
their $\lambda_{n}$ and $\lambda_{c}$ always stay at their initial value, i.e.
one and $\lambda_{c0}$, respectively. b) and c) Evolution of the stretches for
chains parallel and perpendicular to uniaxial load.

Finally, we intend to show how anisotropy is induced in an initially isotropic
material by means of differential damage in different directions. To this aim,
in Fig.5a we show the distribution of the permanent set $\lambda_{c}$ and
$\lambda_{n}$ with respect to latitude. Filaments with a latitude smaller than
$\approx 30^{\circ}$ with respect to the horizontal result undamaged, i.e.
their $\lambda_{n}$ and $\lambda_{c}$ always stay at their undamaged value. In
particular Fig.5b and Fig.5c refer to chains parallel and orthogonal to the
uniaxial load, respectively. The first ones accumulate damage during growing
cyclic loading, whereas the second ones result unaffected, since
$\lambda_{\text{max}}$ is never attained.

Finally to describe the effectiveness of the model in describing the
experimental behavior of different suture materials, in the Appendix we
successfully predict the experimental behavior of sutures reported in [8].

##  4 Conclusions

We propose a predictive model for suture materials. The constitutive
assumptions extend our previous model in [1], where a WLC energy term
introducing variable natural and contour lengths was proposed. As well known
(e.g. see [15]), soft macromolecular materials undergo anisotropic damage.
This was already noticed in [28] and plays a crucial role in the observed
residual stretches. Here we consistently describe such induced anisotropy by
means of the classical microsphere model proposed in [5] in the field of
rubber elasticity. In this vein, an isotropic three-dimensional distribution
of chains is considered. Moreover we introduce the constraint arising from the
network and the resulting entropy variation by considering a new energy term
due to the filament interaction with surrounding forest chains. Finally, our
approach takes into account the relevant unfolding $\rightleftarrows$
refolding phenomena of crystalline regions composing biodegradable copolymers.
As a consequence our approach is able to describe hysteretic behaviors,
permanent damage (Mullins effect [29, 30]) and induced anisotropy. We remark
that our model bridges the micro-to-macro scale by means of the classical
affinity hypothesis diffusely adopted in rubberlike elasticity.

We argue that our constitutive model can be inspiring in the perspective of
engineering design of new biomedical materials, tailored for innovative
technological applications.

As a particular case, we show the effectiveness of our model by considering
the experimental results on uniaxial cyclic tests that we performed on a
biodegradable copolymer (poliglecaprone 25) used for suture threads. The
model, describing all the effects reported above, depends upon only five
microscopically-based parameters with a clear physical interpretation.
Interestingly this feature of the model helps us determining a reasonable
domain of parameters variation in view of the identification procedure. To
this end we carry out a standard optimization procedure (a hybrid particle
swarm method refined by a standard local minimization). The model exhibits a
very efficient capacity in reproducing experimental results with a global
error of 2.62%.

While to describe in details all the obtained results in this paper we
restricted our attention to poliglecaprone 25 sutures, we remark the present
model generality will be useful in describing the mechanical response of
polymers with similar microstructure of interest in material science and
biomedical applications. Such extensions will be the subject of our future
studies.

## Acknowledgments

G. Vitucci is supported by the POR Puglia FESR-FSE project REFIN A1004.22, F.
Trentadue and D. De Tommasi by the Prin Project 2017J4EAYB, G. Puglisi by the
Prin Project 2017KL4EF3 and Gruppo Nazionale per la Fisica Matematica (GNFM)
of the Istituto Nazionale di Alta Matematica (INdAM).

## References

  * [1] F. Trentadue, D. De Tommasi, and G. Puglisi, “A predictive micromechanically-based model for damage and permanent deformations in copolymer sutures,” Journal of the Mechanical Behavior of Biomedical Materials, vol. 115, p. 104277, 2021.
  * [2] T. Rey, G. Chagnon, J.-B. Le Cam, and D. Favier, “Influence of the temperature on the mechanical behaviour of filled and unfilled silicone rubbers,” Polymer Testing, vol. 32, no. 3, pp. 492–501, 2013.
  * [3] J. S. Martinez, J.-B. Le Cam, X. Balandraud, E. Toussaint, and J. Caillard, “Mechanisms of deformation in crystallizable natural rubber. part 1: thermal characterization,” Polymer, vol. 54, no. 11, pp. 2717–2726, 2013.
  * [4] M. Doi, S. F. Edwards, and S. F. Edwards, The theory of polymer dynamics, vol. 73.  oxford university press, 1988.
  * [5] S. Göktepe and C. Miehe, “A micro–macro approach to rubber-like materials. part iii: The micro-sphere model of anisotropic mullins-type damage,” Journal of the Mechanics and Physics of Solids, vol. 53, no. 10, pp. 2259–2283, 2005.
  * [6] P. Bažant and B. Oh, “Efficient numerical integration on the surface of a sphere,” ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 66, no. 1, pp. 37–49, 1986.
  * [7] R. S. Bezwada, D. D. Jamiolkowski, I.-Y. Lee, V. Agarwal, J. Persivale, S. Trenka-Benthin, M. Erneta, J. Suryadevara, A. Yang, and S. Liu, “Monocryl® suture, a new ultra-pliable absorbable monofilament suture,” Biomaterials, vol. 16, no. 15, pp. 1141–1148, 1995\.
  * [8] A. Elías-Zúñiga, B. Montoya, W. Ortega-Lara, E. Flores-Villalba, C. A. Rodríguez, H. R. Siller, J. A. Díaz-Elizondo, and O. Martínez-Romero, “Stress-softening and residual strain effects in suture materials,” Advances in Materials Science and Engineering, vol. 2013, 2013.
  * [9] A. Elías-Zúñiga, K. Baylón, I. Ferrer, L. Serenó, M. L. Garcia-Romeu, I. Bagudanch, J. Grabalosa, T. Pérez-Recio, O. Martínez-Romero, W. Ortega-Lara, et al., “On the rule of mixtures for predicting stress-softening and residual strain effects in biological tissues and biocompatible materials,” Materials, vol. 7, no. 1, pp. 441–456, 2014.
  * [10] M. Rubinstein, R. H. Colby, et al., Polymer physics, vol. 23.  Oxford university press New York, 2003.
  * [11] C. Miehe, S. Göktepe, and F. Lulei, “A micro-macro approach to rubber-like materials—part i: the non-affine micro-sphere model of rubber elasticity,” Journal of the Mechanics and Physics of Solids, vol. 52, no. 11, pp. 2617–2660, 2004.
  * [12] C. Miehe and S. Göktepe, “A micro–macro approach to rubber-like materials. part ii: The micro-sphere model of finite rubber viscoelasticity,” Journal of the Mechanics and Physics of Solids, vol. 53, no. 10, pp. 2231–2258, 2005.
  * [13] D. De Tommasi, N. Millardi, G. Puglisi, and G. Saccomandi, “An energetic model for macromolecules unfolding in stretching experiments,” Journal of The Royal Society Interface, vol. 10, no. 88, p. 20130651, 2013.
  * [14] D. De Tommasi, G. Puglisi, and G. Saccomandi, “Multiscale mechanics of macromolecular materials with unfolding domains,” Journal of the Mechanics and Physics of Solids, vol. 78, pp. 154–172, 2015.
  * [15] G. Puglisi, D. De Tommasi, M. Pantano, N. Pugno, and G. Saccomandi, “Micromechanical model for protein materials: From macromolecules to macroscopic fibers,” Physical Review E, vol. 96, no. 4, p. 042407, 2017\.
  * [16] D. De Tommasi, G. Puglisi, and G. Saccomandi, “Damage, self-healing, and hysteresis in spider silks,” Biophysical journal, vol. 98, no. 9, pp. 1941–1948, 2010.
  * [17] H. J. Qi and M. C. Boyce, “Stress–strain behavior of thermoplastic polyurethanes,” Mechanics of materials, vol. 37, no. 8, pp. 817–839, 2005\.
  * [18] F. Yeh, B. S. Hsiao, B. B. Sauer, S. Michel, and H. W. Siesler, “In-situ studies of structure development during deformation of a segmented poly (urethane- urea) elastomer,” Macromolecules, vol. 36, no. 6, pp. 1940–1954, 2003.
  * [19] G. Marckmann, E. Verron, L. Gornet, G. Chagnon, P. Charrier, and P. Fort, “A theory of network alteration for the mullins effect,” Journal of the Mechanics and Physics of Solids, vol. 50, no. 9, pp. 2011–2028, 2002.
  * [20] A. Dorfmann and R. W. Ogden, “A constitutive model for the mullins effect with permanent set in particle-reinforced rubber,” International Journal of Solids and Structures, vol. 41, no. 7, pp. 1855–1878, 2004.
  * [21] J. Diani, M. Brieu, and J. Vacherand, “A damage directional constitutive model for mullins effect with permanent set and induced anisotropy,” European Journal of Mechanics-A/Solids, vol. 25, no. 3, pp. 483–496, 2006.
  * [22] G. Machado, G. Chagnon, and D. Favier, “Induced anisotropy by the mullins effect in filled silicone rubber,” Mechanics of Materials, vol. 50, pp. 70–80, 2012.
  * [23] W. Kuhn and F. Grün, “Beziehungen zwischen elastischen konstanten und dehnungsdoppelbrechung hochelastischer stoffe,” Kolloid-Zeitschrift, vol. 101, no. 3, pp. 248–271, 1942.
  * [24] G. E. Fantner, E. Oroudjev, G. Schitter, L. S. Golde, P. Thurner, M. M. Finch, P. Turner, T. Gutsmann, D. E. Morse, H. Hansma, et al., “Sacrificial bonds and hidden length: unraveling molecular mesostructures in tough materials,” Biophysical journal, vol. 90, no. 4, pp. 1411–1418, 2006.
  * [25] J. F. Marko and E. D. Siggia, “Stretching dna,” Macromolecules, vol. 28, no. 26, pp. 8759–8770, 1995.
  * [26] B. D. Coleman and M. E. Gurtin, “Thermodynamics with internal state variables,” The journal of chemical physics, vol. 47, no. 2, pp. 597–613, 1967.
  * [27] G. Bertotti, Hysteresis in magnetism: for physicists, materials scientists, and engineers.  Gulf Professional Publishing, 1998.
  * [28] L. Mullins, “Permanent set in vulcanized rubber,” Rubber Chemistry and Technology, vol. 22, no. 4, pp. 1036–1044, 1949.
  * [29] L. Mullins, “Softening of rubber by deformation,” Rubber chemistry and technology, vol. 42, no. 1, pp. 339–362, 1969.
  * [30] J. Diani, B. Fayolle, and P. Gilormini, “A review on the mullins effect,” European Polymer Journal, vol. 45, no. 3, pp. 601–612, 2009.

##  Appendix A Comparison with existing experimental tests

Figure 6: Comparison between modified theory and literature experiments. The evolution law includes the parameter $\beta_{u1}$ defined in Eq.26. Uniaxial cyclic tests on four absorbable suture threads carried out by [8]. Data are shown in black, theory including Eq.26 is shown in red. Materials: a) Monoplus; b) Monosyn; c) Polydioxanone 2.0; d) Polydioxanone 4.0. |  $\kappa$[$10^{8}\times$Pa] |  $\nu$[$10^{8}\times$Pa] | $\lambda_{c0}$ | $\beta_{u0}$ | $\beta_{u1}$ | err[%]
---|---|---|---|---|---|---
Monoplus | 3.421 | 1.499 | 1.339 | -0.012 | -0.091 | 3.94
Monosyn | 1.023 | -0.130 | 1.201 | -0.986 | 0.068 | 6.77
Polydioxanone 2.0 | 3.450 | -1.298 | 1.3107 | 0.557 | -0.688 | 2.07
Polydioxanone 4.0 | 0.453 | -6.010 | 1.064 | -0.232 | 0.395 | 3.04
Table 2: Parameters of the fit with $\beta_{u1}$ defined in Eq.26.

In this Section we briefly explore the performance of our model on four other
absorbable suture threads experimental data as published by [8]. In order to
adapt our formulation to this larger class of materials, we introduce an
additional parameter $\beta_{u1}$ as follows:

$\lambda_{n}=\sqrt{\lambda_{\text{max}}}\left(\frac{\lambda}{\lambda_{\text{max}}}\right)^{{\beta_{u}+\beta_{u1}\lambda+\beta_{r}\sqrt{\lambda-\lambda_{\text{min}}}}}\geq\bar{\lambda}_{n}\quad.$
(26)

The results of the parameter identification procedure are shown in Fig.2 and
Table 2. Notice that the lack of reloading data does not allow estimating the
reloading parameter $\beta_{r}$ so that for this set of experiments we only
need five parameters even including $\beta_{u1}$.

Notice that the model of [8, 9] required seven parameters, it did not include
internal hysteresis and did not produce a better match than the present work.

The errors of the fit result extremely low and this makes us confident that
the developed multiscale model is physically sound and versatile in the
perspective of describing the mechanical behavior of a vast class of polymeric
materials of biomedical interest. Further analysis will be the subject of
future work.