Bioinspired sutured materials for strength and toughness

16 Pages·2017·5.54 MB·English

by  Idris A. Malik

International Journal of Solids and Structures 138 (2018) 118–133
Contents lists available at ScienceDirect
International Journal of Solids and Structures

journal homepage: www.elsevier.com/locate/ijsolstr
Bioinspired sutured materials for strength and toughness: Pullout

mechanisms and geometric enrichments

Idris A. Malik, Francois Barthelat ∗
Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, QC H3A 2K6, Canada
a r t i c l e i n f o a b s t r a c t
Article history: Hard structural elements in nature are often joined with sutures lines, as seen in human skull,
Received 9 October 2017 cephalopods or turtle shell. These sutures can arrest cracks, and can provide flexibility for respiration,
Revised 24 December 2017 locomotion or growth. In this paper we introduce a morphometric method to capture the complex shape
Available online 6 January 2018
of sutured interfaces using only a few parameters. The method is simple, and can capture relatively com-
Keywords: plex suture geometries with re-entrants, interlocking features. The study starts with a simple jigsaw-like
Sutured lines model which is enriched with additional features (plateau regions in dovetail-like sutures, multiple lock-
Shape descriptors ing sites). For each case, closed form and finite elements solutions are developed to capture the full non-
Analytical models linear pullout response and to predict the maximum stress (and potential fracture) in the solid material.
Finite element models These models were then used to identify the geometries and interface properties (friction) that lead to
Optimization optimum combinations of strength and energy absorption. Suture designs that reduced frictional stress
with low coefficient of friction or with multiple contact points were the most efficient. The results can
serve as guidelines to design and optimization of non-adhesive sutures with arbitrary shapes made of arc
of circles and lines. We found that the best designs involve low coefficient of friction, which raises an
interesting hypothesis on the function of the protein layer in natural sutured lines: This soft layer could
act as “lubricant”to prevent the fracture of the solid structures.
©2018 Elsevier Ltd. All rights reserved.
1. Introduction digitating suture lines in human skull which become more com-
plicated from infant to adult ( Miura et al., 2009; Maloul et al., ).
Components made of hard materials can be joined by suture Some of these interfaces form anti-trapezoidal sutures which are
lines, a structural feature found in multiple examples in nature: interlocked as seen in linking the girdles of diatoms ( Genkal and
shells of cephalopods ( Allen, 2007 ), human skull ( Coats and Mar- Popovskaya, 2008; Manoylov et al., 2009 ), while others display si-
gulies, 2006; Maloul et al., 2014; Miura et al., 2009 ), carapace of nusoidal interfaces which increases resistance to crack propagation
turtle ( Achrai et al., 2014; Chen et al., 2015; Zhang et al., 2012 ), ( Li et al., 2011 ).
wood peckers beak ( Lee et al., 2014 ) ( Fig. 1 ), and in many other The softer interface materials at the sutures enable the rela-
hard biological materials where weak interfaces govern deforma- tive displacement and/or relative rotation of harder structural com-
tion and fracture mechanisms ( Dunlop et al., 2011; Barthelat et al., ponents, which facilitates locomotion, respiration or growth ( Lin
2016 ). In these examples, stiff skeletal material (mineralized pro- et al., 2014; Li et al., 2011 ). Sutured interfaces can also absorb im-
teinaceous matrices, keratin) are joined by thin lines of interfa- pact energy ( Lee et al., 2014 ), channel the propagation of cracks
cial material which are much softer. The geometrical complex- into toughening configurations, or act as a source of local defor-
ity of the suture lines ranges from nearly straight sutured inter- mation that can spread energy dissipative mechanisms throughout
faces in new born baby skull ( Coats and Margulies, 2006; Miura large volumes ( Barthelat et al., 2007; Fratzl et al., 2004 ). The geom-
et al., 2009 ), to more complex ceratitic and ammonitic sutures etry of sutured lines largely governs their mechanical response ( Li
with fractals geometries ( Allen, 2007; Li et al., 2012; Lin et al., et al., 2012, 2013; Lin et al., 2014, 2014; Zavattieri et al., 2008 ). In
2014 ). In cephalopods such as ammonoid, which produce angu- particular, interlocking geometrical features can increase strength
lar or dendritic sutures, the complexity of the suture lines varies and energy dissipation ( Malik and Barthelat, 2016; Mirkhalaf et al.,
across species ( Allen, 2007 ), or with growth as seen in the inter- 2014; Mirkhalaf and Barthelat, 2017; Haldar et al., 2017 ). We re-
cently developed analytical and finite element models that cap-
ture the complete nonlinear pullout response of sutures with sim-
ple interlocking jigsaw-like geometries, based on frictional con-
∗ Corresponding author.
E-mail address: [email protected] (F. Barthelat). tact and linear elasticity ( Malik et al., 2017 ). These models demon-
https://doi.org/10.1016/j.ijsolstr.2018.01.004
0020-7683/©2018ElsevierLtd.Allrightsreserved.
I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133 119
Fig. 1. Examples of sutured interfaces in nature: (a) Ammonite shell ( Ceratitic ammonoid ) with intricate suture lines ( Lin et al., 2014 ), (b) Pan troglodytes cranial sutures
(adapted from Cray et al., 2010 ), (c) Osteoderms of a leatherback sea turtle shell (adapted from Chen et al., 2015 ), (d) red-bellied woodpecker ( Melanerpes carolinus ) beak
(adapted from Lee et al., 2014 ).
Fig. 2. (a) A parametric ( φ-s ) curve showing the cumulative angular function as function of curvilinear position; (b) Suture profile in ( x,y ) reconstructed from the ( φ-s )
function.
φ
strated how geometry and local friction coefficient govern stiffness, only considered ( s ) functions that are multilinear ( Fig. 2 a). The
φ
strength and energy absorption, and how geometry and friction 2D profile of the suture can be reconstructed from the ( s ) func-
can be tuned to optimize the mechanical response. In this arti- tion in the x-y coordinate system using:
csilde ewr es upturereses nwt inthe wm eoxrete ncosimonpsle xto mthoerpseh omloogdieesls: , dwovheetraei l-wliek ec osun-- ⎧⎪⎨ x (s ) = (cid:6) s cos ( φ(s ) )d s
ttuyprees oafn ds ustuutruer ews ew pitehr ftowrmo eodr mano reex phaaiurss toivfe l oscekairncgh stitoe sd. eFtoerr meainche ⎪⎩ y (s ) = (cid:6)0 s sin ( φ(s ) )d s (1)
the design parameters which maximize strength and energy ab- 0
sorption. A comparative study of the different suture morphologies Where the point x = 0 and y = 0 coincides with s = 0. This pro-
is provided at the end of the discussion.
cess generates a periodic unit cell of the suture ( Fig. 2 b).
For this study we required the contour ( x-y ) of the suture to be
2. Overview of the suture geometry periodic and continuous. We also required contours with no sharp
corner or kinks which would generate stress concentrations and
φ
Capturing the two-dimensional geometry of a curved suture lead to sub-optimal designs. The function ( s ) therefore had to be
line requires robust, high fidelity yet relatively simple mathemati- continuous. In addition, for a suture line whose general orientation
φ
cal models. Methods used in the past include descriptive methods, is aligned with the axis x , the local tangential angle must take a
pattern-matching using geographic information systems, complex- zero value at least once within the periodic unit cell. For simplic-
ity indices, and morphometric methods ( Allen, 2007; Manoylov ity we chose φ(0) = 0. In this work we also only considered sutures
et al., 2009; Saunders et al., 1999 ). In this work, we used a sim- with a symmetry about line x = w /2, where w is the width of the
ple 2D shape descriptor approach similar to the morphometric unit cell in the x-y space. This symmetry implies that the func-
method. Shape descriptors are mathematical objects which can tion φ( s ) is antisymmetric about the line s = 2 s 0 ( Fig. 2 a). Finally,
captures geometrical features in a simplified and condensed fash- we sought optimum geometries for the suture and therefore we
ion, for example the radius of a circle, the surface of an area, or focused on geometries that produces identical stresses on either
geometrical eccentricity ( Fang et al., 2015; Kim and Kim, 20 0 0 ). side of the suture line, because asymmetric sutures lines would be
The shape descriptor we used in this study is based on a cumula- sub-optimal with one side inevitably be “better” than the other.
φ
tive angular function of the contour of the suture as function of Therefore we only considered suture lines with a 180 ° rotational
φ
the curvilinear position s along the contour ( Fig. 2 ). The approach symmetry, which implies that the function ( s ) is symmetric about
is versatile and more importantly, it enables the modelling of su- the line s = s 0 ( Fig. 2 a). Considering these symmetries, the func-
φ
ture lines with re-entrant, interlocking features. For this study we tions ( s ) only need to be defined over a quarter of the curvilinear
120 I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133
are given by:
⎧⎪⎨ w = 2 (cid:6) 2 s 0 cos ( φ(s ) )d s
⎪⎩ L = (cid:6) 2 ss 0 = s0i n ( φ(s ) )d s (2)
s =0
This “shape generating algorithm”has a few properties that are
useful to outline here. The local radius of curvature R on the ( x-y )
profile of the suture is related to the local suture angle by Rd φ= ds ,
φ
so that the local slope of ( s ) is:
φ
d 1
= (3)
ds R
An important implication is that horizontal segments in the
φ( s )function ( φ’ ( s ) = 0) correspond to straight segments on the ( x-
y ) profile ( R = ∞ ). This shape descriptor approach was used to gen-
erate a wide array of suture geometries, and the pullout mechan-
ics of each of these geometries was captured with the model de-
scribed below.
Fig. 3. (a) Half-unit cell for the suture with symmetry and boundary conditions
applied. (b) Free body diagram of the half-unit cell; (c) free body diagram of the
lower tab only, exposing the normal and frictional forces transmitted at the contact 3. Pullout models
between the tabs.
In terms of mechanics, this study focused on the full pullout re-
sponse of the suture along the y direction ( Fig. 3 ). We only consid-
length of the contour in the periodic cell (0 ≤ s ≤ s 0 , Fig. 2 ). The ered sutures with no adhesive at the interfaces, so that the pullout
first quarter of the ( x - y ) contour was reconstructed using Eq. (1) up response was governed only by contact mechanics, friction and ge-
to s = s 0 , and a 180 ° rotational symmetry was then applied about ometric interlocking. The solid part of the suture was assumed νto
points x ( s 0 ), y ( s 0 ), in order to produce a half unit cell which was be isotropic and linear elastic (modulus E and Poisson’s ratio ),
used for the analysis of kinematics, forces and stresses. For dis- and 2D plane stress conditions were used. Fig. 3 a shows a repre-
play purposes, the full unit cell may be reconstructed with a sim- sentative volume element (RVE) of the suture geometry. The pull-
ple symmetry ( Fig. 2 b). The full unit cell has width w , and height out was simulated using a displacement controlled boundary con-
L + 2 h where L is the projected length of the suture on the y axis, ditions:
(cid:7)
and h is the height of two regions included in the model on either u y (x, −h ) = 0
sides of the suture. The width, w and the height, L of the suture (4)
u y (x, L + h ) = u
Fig. 4. (a) ( φ-s ) curve for a one-parameter suture; (b) Corresponding profile; (c) model with symmetry and boundary conditions; (d) The “strength” of the geometrical
interlocking is governed by the interlocking angle θ0 , where θ0 < 60 °to prevent the re-entrant regions of the tabs to intersect (case highlighted in red). (For interpretation
of the references to colour in this figure legend, the reader is referred to the web version of this article.)
I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133 121
Fig. 5. Effects of (a) friction f and (b) interlocking angle θ0 on the pullout response of the suture; effects of (c) friction f and (d) interlocking angle θ0 on maximum tensile
stress in the solid tabs.
Where h is the height of the upper and lower portion from the unit cell is zero, the compressive forces in the suture region must
suture interface ( Fig. 2 c). The sides of the model were subjected to be balanced by tensile forces F T transmitted on either sides of the
the periodic boundary conditions: suture line ( Fig. 3 b). Fig. 3 c shows a free body diagram of the lower
(cid:7)
u x (w/ 2 , y ) −u x (−w/ 2 , y ) = w ε¯ x section of the suture, which exposes the contact forces transmit-
(5) ted at the interfaces. These forces can be decomposed into nor-
u y (w/ 2 , y ) = u y (−w/ 2 , y ) mal force(s) P i and frictional force(s) fP i (only one contact force is
ε showed on Fig. 3 c, but there might be more sets of contact forces
Where ¯ x is the average strain in the x (transverse) direction.
depending on the design of the suture). These contact forces will
The geometry and loading conditions are symmetric about the y -
translate into a pullout force through the equation:
axis so that:
u x (x, y ) = −u x (−x, y ) (6) F = 2 (cid:8)N P i ( sin θ+ f cos θ) (9)
Combining Eqs. (5) and (6) , the periodic boundary conditions i =1
are written:
(cid:7)
u x (0 , y ) = 0 soliTdh ep asretst ooff ftohrec essu tsuhroew wn hiinc hF img. u3s t ablseo mgeonneitroartee ds tbreescsaeuss ei nf rtahce-
(7)
u x ( w/ 2, y ) = w2 ε¯ x turing the material would abort the pullout mechanisms and can-
cel their potential benefits. Experiments and stress analysis have
In cases where h is sufficiently large, the stiffness of the solid
shown that for brittle material in sliding dry contact, the highest
regions in the transverse direction is high enough to neglect
any deformation in the transverse direction, i.e. ε¯ x = 0 . Equations ttehnes icloen sttarcets saerse ao c(c Muar liakt etht ea ls.,u r2f0a1c7e ).o fF othr ee ascuhtu oref , tahte thgeeo emdegter ieosf
(7) then become ( Fig. 3 a):
(cid:7) presented below, the maximum tensile stress in the solid part was
u x (0 , y ) = 0 calculated for the entire pullout sequence.
u x (w/ 2 , y ) = 0 (8) binaWtieo na lsoof ufisneidte neulmemereincta l smimeuthlaotdiosn fso r( AthNeS YmS opdaerlas,m wetitrhic ad ecosimgn-
φ
When the upper suture is pulled along the y axis, the interlock- language) interfaced with Matlab (R2016a, MA, US). (s) curves
ing at the suture generates a pullout force F along the y axis ( F/2 were generated and automatically transformed into suture geome-
on the half unit cell, Fig. 3 b). The contact forces at the interfaces tries into x-y profiles using a Matlab. The same code was used
also generate a horizontal component, giving rise to a compressive to automatically generate APDL input files for ANSYS. The mod-
forces F c along the x axis. Because the net horizontal force on the els were meshed with quadratic, plane stress element (PLANE 183),
122 I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133
Fig. 6. (a) and (b) contour plots of maximum pressure (minimum principal stress) for θ0 = 5 °and f = 0.4; (c) contour plot of maximum principal stress (maximum tensile
stress in the tabs); (d) and (e) traction and maximum tensile stresses as function of pullout distance showing a good agreement between the analytical and finite elements
results (extracted from Malik et al., 2017 ).
and contact elements (CONTA 172, TARGE 169, symmetric contact) locking sutures. Within each type of geometry we sought the geo-
were used to simulate sliding and frictional contact at the inter- metrical parameters and materials properties that lead to optimum
face. To appropriately mesh the model we used an adaptive mesh- combinations of high strength and high energy absorption, while
ing approach for each geometry. A first model was run with a uni- preventing the brittle fracture of the solid part of the suture.
form mesh, and the mesh was automatically refined at the regions
of highest stresses. The procedure was repeated automatically until
4.1. Single jigsaw sutures
the results for pullout response and maximum tensile stress in the
solid converged, ensuring mesh independent results. Typically the
The single jigsaw sutures are the simplest form of geometric
size of the elements in the converged mesh was about R /50 0 0 in θ
parameters, with a radius of curvature R 0 , and a locking angle 0
the contact region, where R is the radius of curvature. The number
([27], Fig. 4 ). Since the mechanisms captured here have no specific
of time steps in simulation was also adapted automatically to en-
length scale, all results are normalized by the size of the model,
sure converged results, typically about 500. The results were auto- θ
leaving 0 the only geometrical parameter. The other geometric pa-
matically post-processed using ANSYS ADPL and Matlab. This fully
rameters can be found using Eqs. (2) and (4) :
automated exploration procedure enabled the evaluation of thou- ⎧
sands of models with different suture geometries and coefficients ⎪⎨ φ0 = θ0 + π/ 2
of friction. ⎪w = 4 R 0 sin φ0 (10)
⎩
4. Exploration of suture geometries L = 2 R 0 (1 −cos φ0 )
The length of the 1/4 contour of the suture is s 0 = R 0 φ0 and the
In this section we explore the effect of several suture geome- φ( s ) function is simply ( Fig. 4 a):
tries on pullout behavior. We organized the exploration by starting
womiteht rai csailm ppalrea mdeestiegrn. W(sein tghleen j ipgrsoawgr essustivuerelys )e bnariscehde do nt hae sgienogmlee gtrey- φ( s ) = Rs 0 , 0 ≤s ≤s 0
of the suture by adding more geometrical features through addi- Fig. 4 b shows the reconstructed profile of the suture, which
tional parameters: dovetail-like sutures, double locking sutures, N - consists of four appended arcs of circle. The level of interlocking
I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133 123
Fig. 7. (a) Possible combinations of the parameters θ0 and f for optimum designs; (b) Optimum energy absorbed and optimum pullout strength calculated using the con-
straint σs / E = 1/100.
θ
increases when 0 increases as shown on Fig. 4 d, up to the ex-
treme case θ0 = 60 ° where the suture line intersects which corre-
sponds to the maximum geometrically allowed locking angle. Su-
tures with θ0 ≥ 60 ° were therefore excluded from the exploration
as physically inacceptable. An analytical solution for the pullout
response based on simple kinematics and contact mechanics was
presented in a previous article ( Malik et al., 2017 ), with the main
results are summarized below. Fig. 4 b and c give the kinematics
relations ( Malik et al., 2017 ):
( 2 R 0 −δ) cos θ = 2 R 0 cos θ0 (11)
u = 2 R 0 sin θ0 −( 2 R 0 −δ) sin θ (12)
Using contact mechanics, the non-dimensional interference is
given as ( Johnson and Johnson, 1987 ):
(cid:9) (cid:10) (cid:11) (cid:12)
δ
= π2 P ln 4 πR 0 tE −1 (13)
R 0 R 0 tE P
Where t is the thickness of the tab. This equation is solved nu-
merically to determine the non-dimensional contact force R 0P t E and
the non-dimensional pullout force from Eq. (9) is given as:
(cid:13) (cid:14)
F P sin θ+ f cos θ
= θ (14)
wtE R 0 tE 2 cos 0
The effect of Poisson’s effects are neglected in this solution, but
finite elements confirm that Poisson’s ratio has little effects on the
solutions.
Fig. 5 a and b show the effects of the friction coefficient f , and
θ
interlocking angle 0 on the pullout response of the suture. High
coefficients of friction f lead to relatively high strength because of
increased friction at the contact point, but do not change the max-
imum pullout distance which is governed by geometric parame-
ters. Higher interlocking angles θ0 increase the strength because of Fig. 8. (a) ( φ-s ) curve for a two-parameter suture; (b) Corresponding profile; (c)
increased geometrical interlocking ( Fig. 5 b), and also increase the The “strength”of the geometrical interlocking is governed by the interlocking angle
maximum pullout distance because the tabs stay in contact over θ0 and the plateau length d/R 0 . The geometries highlighted in red are not physically
θ acceptable. (For interpretation of the references to color in this figure legend, the
a longer pullout distance. The friction f and interlocking angle 0 reader is referred to the web version of this article.)
have therefore positive effects on strength and energy absorption.
However increasing these two parameters also increase frictional
stresses, which can lead to the fracture of the tabs. The maximum illustrated with finite element results, with the lowest principal
tceonnstialect sstrteressss iens t(h Me asluikt uerte aisl. ,d 2iv0i1d7e )d, winhtoic hfr iccatino nblee ses vaanluda ftreidct iforonmal setsrte stes nsσimEle in (sit.ere. sms aσxmEia mx (u Fmig . p6r ce)s. sTuhree, cFoing.t a6c at parneds sub)r ea insd d tishter ibhuigtehd-
contact solutions. These frictional stresses produce the maximum over the contact following the expected parabolic profile. The ten-
tensile stresses in the solid tabs, which are plotted as function of sile stress is maximum at the edge of the contact surface which is
pullout distance for different coefficient of friction ( Fig. 5 c) and dif- “behind” the direction of sliding. Fig. 6 d and e show the pullout
ferent locking angles ( Fig. 5 d). The contact stresses can also be (force-displacement) curve and the maximum local stress as func-
124 I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133
Fig. 9. Stages of the pullout for a dovetail suture; (a) Initial stage (no force applied), (b) first pullout stage where the flat faces are in contact, (c) second stage where only
the rounded sections are in contact.
tion of pullout distance. The analytical solution and finite element this design: the maximum strength of the suture is only about 5%
models are in good agreement. of the tensile strength of the solid material. In the upcoming sec-
θ
The interlocking angle 0 , and friction coefficient f , increase the tions we explore enriched geometries that aim at reducing the fric-
maximum pulling force and total energy absorbed during pullout. tional contact stresses by distributing the contact over larger areas.
However the associated local stresses in the solid may cause a pre-
mature fracture of the tab, therefore there is need for optimiza-
4.2. Dovetail-like sutures
tion. Here we performed an exhaustive search of the design space
to identify the best combination(s) of design parameters for any
As a first extension to the simple jigsaw geometry, we increased
given set of desired of normalized stiffness, strength, maximum ex-
the contact area in order to better distribute and to decrease the
tension, and energy absorption. The limiting factor for the design
contact stresses, and in particular those associated with friction.
is the fracture of the solid tabs, which is governed by the strength
Reducing the contact stresses can delay the fracture of the solid
of the material. The tabs will therefore fracture when σmax / E = σs materials, which can enable more extreme locking geometries and
σ
/ E , where max is the maximum tensile stress predicted from the more efficient designs. A straight region of length d was added
σ
model and s is the tensile strength of the solid material. Here on the suture contour, which produced a “dovetail like” geometry
we present results for σs / E = 1/100 , which is a common ratio for ( Fig. 8 a,b). This dovetail suture can be described with three inde-
engineering materials ( Ashby, 2011 ). For a given value of friction θ
pendent parameters: interlocking angle 0 , radius of curvature R 0
coefficient, it is therefore possible to identify the optimum locking
and length of straight segment d , which were reduced to two pa-
angle that will be the largest while preventing fracture of the tabs. θ
The strength is simply Fw mtaEx , the maximum elongation is u mLa x and rraammeetteerrss a( re0 gainvde nd /b Ry 0: ) after normalization. Other geometrical pa-
the energy absorbed is given by the area under the pullout force- ⎧
dθi0s pawlnUtaL dcEe m=f te(cid:15)hn0 ua t mt a cxwu wrivFlt lEe ,dp Lur e. vFeign. t7 t aa sbh ofrwacst uthree aalnlodw eanbsleu rceo mthbei ncaotmiopnlse otef ⎪⎪⎪⎪⎨ φs 00 / R= 0 θ=0 +φ0 π +/ 2d / 2 R 0 (15)
pcornue laltoshuee t wltihintehe sdθue0t fiuanrene.dd S fib,n yct heσ es mtoraepx nti/gm Et h=u m σan s cdo/ Eem =nbe 1irn/g1ay0t i0ao.bn ssAo sro pfe txiθop0ne cabtneoddt h, f iiflni e-f ⎪⎪⎪⎪⎩ wL// R R 0 0 == 2 4( 1si n− φc0o s+ φ 20( )d +/ R ( 0d )/ c Ro 0s ) φsi0n φ0
θ
is increased then 0 must be decreased to prevent the fracture of The φ( s ) function ( Fig. 8 a) was defined as:
the tabs. Fig. 7 b shows the corresponding optimal energy absorp- (cid:7)
tpfriuoiclnlto iou(n tw UtcsL toEr e)effi onpgtc tiheann itds fa t=chh e0i ,e oavpnetddi m awta ilat h sv taarl eulnoegc ktFh min ag(x /Fwwa mtnatExEg ) le o∼p tθ 0. 0.5T =h × e1 120m.7 −5a3 °x . iaHmnoudwm a- φφ(( ss )) == φRs 0 0 , , 0s 0 ≤ −sd ≤/ 2s 0≤ −s d≤/ 2s 0 (16)
ever, this design does not dissipate any energy upon pullout, be- Fig. 8 c shows a set of dovetail suture profiles obtained for dif-
cthaeu shei gthhee sftr iecntieorngayl ddiissssiippaattiiovne pmoescshibalnei sf em isu satb bseen itn c( fr =ea 0se).d T, oa nreda θch0 fgeeroemnte tcroym ibs inreactoiovnesr eodf. θW0 itahn dd /dR/ 0R > 0 . 0F oar wd/iRd 0e =r a0n tghee osfi nggeloem jiegtsraiews
must be decreased. The bell-shape of this curve shows there is an can be obtained, but some combinations of parameters lead to the
optimum point where the energy dissipation is maximum, which contour intersecting itself. The condition for the contours not to
if s= a 0c.h06ie . vTehdi sw eitxhh aau lsoticvkein sge aarncghl ec aθn0 =th 9e.2re5f °oaren db efr iucstieodn tcoo eidffiecniteinfyt intersect is 2(cid:16) R 0 < w , which(cid:18) c an be wr(cid:13)itten: (cid:14)
tahned /sourt utoreu gghenoemsse. trTyh ea nreds uloltcsa la lfsroic thioignh ltihgahtt wthiell mopatiinm liizmei tsattrieonng tohf θ0 < arccos (cid:17) d 2 R+ 0 4 R 0 2 −arctan 2d R 0 (17)
I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133 125
Fig. 10. Effects of (a) friction f (b) interlocking angle θ0 , and (c) plateau length d/R 0 , on the pullout response of the suture; effects of (d) friction f (e) interlocking angle θ0 ,
and (f) plateau length d/R 0 , on maximum tensile stress in the solid tabs.
On Fig. 8 c, Eq. (17) defines the boundary between admissi- from d to zero. The pullout distance in the first stage is given as:
ble geometries and inadmissible geometries. As the plateau length
d/R 0 , increases from zero (single locking) to higher values, the u 0 = (cid:9)d cos θ0 (21)
θ
range of allowable locking angles 0 becomes narrower. To pre- Where (cid:9)d is the difference in the sliding length as the pullout
dict the pullout response of the dovetail suture we used the same
progresses from zero to d . At the second stage the pullout distance
procedure as for the single jigsaw model. An analytical solution
is the same as that of a single jigsaw which is given as:
was first developed by using a 2D flat punch with rounded edge
as base solution for the contact interaction ( Ciavarella et al., 1998, u 1 = 2 R 0 sin θ0 −( 2 R 0 −δ) sin θ (22)
20 02; Giannakopoulos and Chenut, 20 0 0; Sackfield et al., 20 05 ). where δ is the interference between two disks pressed against
The kinematics of two disks in contact is the same as that de- each other and θ is used to track the evolution of the interlock-
scribed above using Eq. (11 ), and the normalized contact radius ing angle.
a/R 0 can be determined from non-Hertzian contact solution for two Resolving the force components vertically and normalizing by
similar d(cid:10)isks(cid:11) in(cid:19) con(cid:13)tact(cid:10) Eq. (cid:11)(18(cid:14)) ( Joh(cid:20)nson and Johnson, 1987 ). the suture wi(cid:13)dth provides the ave(cid:14)rage pullout force:
Rδ 0 = 12 Ra 0 2 ln 16 Ra 0 2 −1 (18) wFt E = R 0P t E 2 scions θ θ0+ − f dc osisn θ θ0 (23)
The normalized contact surface a/R 0 can be obtained by solving The angle θ is used to track the progressive pullout of the su-
Eq. (18) numerically. From a/R 0 one can compute the normal force ture, and it remains constant along the flat portion of the suture,
for a flat punch with rounded ends for plane stress condition is but later evolves from + θ0 to –θ0 at the rounded ends of the su-
given as ( Ciavarella et al., 2002 ): ture. Fig. 10 a–c show the effect of friction coefficient f , interlock-
(cid:13) (cid:14)2 ⎡ π (cid:23) (cid:13) (cid:14)2 (cid:13) (cid:14)⎤ ing angle θ0 , and plateau length d/R 0 , on the average pullout force.
P 1 b ⎣ d d d ⎦ Friction ( Fig. 10 a) and interlocking angle ( Fig. 10 b) have the same
= − 1 − −arcsin (19)
R 0 tE 2 R 0 2 2 b 2 b 2 b effect as we observed for the single jigsaw design- they both in-
crease maximum pullout force and energy absorption. Increasing
the length d of the plateau increases the area of contact, which
b d a
= + (20) increases the pullout force ( Fig. 10 c).
R 0 2 R 0 R 0 The maximum tensile stress in the suture is divided into fric-
The pullout is divided into two stages as shown in Fig. 9 . The tional contact stress and frictionless (hole in an infinite plate
first stage involves the sliding of the flat portion and the inter- loaded by a frictionless pin in the in-plane direction) maxi-
locking angle remains unchanged while the contact length reduces mum tensile stress. The frictionless maximum stress is given as
126 I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133
Fig. 11. (a) and (b): contour plots of maximum pressure for θ0 = 5 °, d = 0.8 , and f = 0.4; (c) contour plot of maximum principal stress (d) and (e) traction and stresses as
function of pullout distance showing a good agreement between the analytical and finite elements results.
( Ciavarella and Decuzzi, 2001 ): Superposition of these two solutions gives the total stress at the
trailing edge of the contact:
σm(Pa)x = RP 0 t 52 − πν (24) σmax = σm(Pa)x + σm(faPx) = RP 0 t 52 − πν + π4 Pt bf k (27)
The sliding frictional contact stress is given as ( Ciavarella et al., Normalizing this equation gives:
2002 ): σ (cid:9) (cid:10) (cid:11) (cid:12)
max = 1π P ( 5 −υ) + 8 R 0 kf (28)
σm(faPx) = 2 fpk (25) E 2 R 0 tE b
The maximum stress increases with interlocking angle, plateau
Where p is the contact pressure given as p = π2t Pb and k is a length, and friction ( Fig. 10 d, e, f). Fig. 11 shows the minimum
geometrical factor that defines the effect of contact length ratio, principal stress, maximum stress contour plots and the compari-
d/2b on the pressure distribution of flat punch with rounded edges. son between the pullout response and maximum stress. The result
If d = 0 , then k = 1 which reduces to a cylinder-on-cylinder contact shows that both the analytical and finite element models are in
( Ciavarella et al., 2002 ). good agreement. For the rest of this section we used the analytical
model because of its simplicity.
p = π2 P To identify optimum sets of design parameters, we followed the
(cid:23)tb same procedure as for the single jigsaw, by performing an exhaus-
θ
k = 1 −( 2 /π) arcsin ( d/ 2 b ) (cid:17) toipvtei mpaurmam seettrsi co fs tpuadraym oent ef ,r s t0h aant dp rde/vRe 0n . tI nfr apcatrutirceu olafr t, hwee t aidbes,n utisfiinedg
1 −( 2 /π) arcsin ( d/ 2 b ) −( 2 /π) ( d/ 2 b ) 1 −( d/ 2 b )2 σs / E = 1/100 for the tensile strength of the solid material. Fig. 12 (a)
θ
σm(faPx) = π4 Pt bf k (26) stihoonw, sa nddif ffeorre nfotu cro dmifbfienraetnito nvsa loufe sf aonf dd /R0 0 . tAhsa te xacpheicetveed , thinisc rceoansidnig-
I.A. Malik, F. Barthelat / International Journal of Solids and Structures 138 (2018) 118–133 127
Fig. 12. (a) Possible combination of the parameters θ0 , f, and d/R 0 for optimum designs; (b) material property map showing the optimum energy absorbed and the optimum
pullout strength for different d/R 0 and for a design constraint σs / E = 1/100.
the plateau d/R 0 must be accompanied by a reduction of f and/or
θ
0 to prevent the fracture of the tab. Fig. 12 b shows the corre-
sponding optimum energy absorption and strength. Compared to
the single jigsaw tab design adding straight regions produced nar-
rower bell-shape curves, because as mentioned above the range of
θ
permissible f and 0 is more restricted. The dovetail design how-
ever produced higher increases energy absorption, up to 2.5 times
higher than what can be obtained from the single jigsaw design.
The maximum pullout strength possible is however the same, at
F max /wtE ∼ 0.5 ×10 −3 for all designs. Interestingly, the optimum
design achieves the highest possible strength and energy absorp-
tion simultaneously. This optimum design has a friction coefficient
of 0.12, a long dovetail d/R o = 0.6 and a vanishingly small lock-
ing angle ( θ0 = 0.125 °). A further increase in the plateau length
( d/R 0 > 0.6 ) did not generate any further improvements.
4.3. Double locking sutures
The dovetail design demonstrated how distributing the contact
stresses over a larger area could delay fracture of the tabs and
lead to higher performance. We now examine another approach,
where the pullout force is transferred over more than one contact
area. "Double locking" suture geometries were obtained by enrich-
φ
ing the (s) with a second segment with a nonzero slope ( Fig. 13 a).
The most interesting cases are produced when this second slope
is negative, which then produces a second locking site of radius
R 1 ( Fig. 13 b). The slope of that segment is therefore −1/R 1 on the
φ
(s) . This enriched design has therefore three independent non-
θ
dimensional geometrical parameters: (i) interlocking angle 0 , (ii)
φ φ
radii ratio R 0 /R 1 , (iii) cumulative angular function 0 / 1 . The su-
ture angle function is written:
(cid:7)φφ(( ss )) == 2Rs 0 φ ,0 −0R s≤ 1 , s ≤s 1s 1≤ s ≤s 0 (29) Fθ(TFih0og e.ra “n1is3ndt.tr eeb(rnyapg )rRt eh( 0t φ ”/aR-to 1isof ). ntTch huoeer fv gegte heofoemom rer eetartf eritcirweaeslon i-chnpeitasge rhratllomoig checkttoieenlrodg r s iiusni nt gu roretevhd;ei sra( nbrfiee)gd Cun boroeyrt r teplheshgepye osinnnicddtae,i lnrltlygho ceapk crirconeefigap ldetaea;nr bg (lcleies).
referred to the web version of this article.)
Other geometrical parameters are written:
⎧
⎪⎪⎪⎪⎪⎪⎨ φφ01 == θ−0θ +1 +π π/ 2/ 2 theT rhaenrgee aoref pgoesosmibelter idc ecsoignnsstr afoinr tsth oisn sRu 0t u/ Rr 1e a. nTdh eθ 0t rawnhsiictiho nr easntrgiclet
⎪⎪⎪⎪⎪⎪⎩ sswL 10 = === 2 2RR [ [R 002 φφ 0 ( (00 R 1 + 0 − + Rc 1Ro ( 1s θ ) 0φ s+0i n) θ −φ1 0)R − 1 (2 c oRs 1 φsi0n − φ1c ]o s φ1 )] (30) θθ11 =ca anr cbceo ws (cid:10)rit12t e(cid:10)nRR: 01 + 1(cid:11) cos θ0 (cid:11) (31)