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Defect sensitivity to failure of highly deformable polymeric materials
Accepted Manuscript Defect sensitivity to failure of highly deformable polymeric materials Roberto Brighenti, Andrea Carpinteri, Federico Artoni PII: DOI: Reference:
S0167-8442(16)30312-3 http://dx.doi.org/10.1016/j.tafmec.2016.12.005 TAFMEC 1794
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
11 October 2016 12 December 2016 27 December 2016
Please cite this article as: R. Brighenti, A. Carpinteri, F. Artoni, Defect sensitivity to failure of highly deformable polymeric materials, Theoretical and Applied Fracture Mechanics (2016), doi: http://dx.doi.org/10.1016/j.tafmec. 2016.12.005
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DEFECT SENSITIVITY TO FAILURE OF HIGHLY DEFORMABLE POLYMERIC MATERIALS
Roberto Brighentia*, Andrea Carpinteria, Federico Artonia a
Dept. of Civil-environmental Engineering & Architecture, Univ. of Parma, Viale delle Scienze 181A, 43124 Parma, ITALY
Abstract The capability of materials to bear loads even in presence of defects like cracks, notches or generic geometric discontinuities is usually indicated as flaw tolerance, and is crucial in modern safety design of structural components. Such a tolerance capability can be remarkable in highly deformable materials (also called soft materials), usually much more pronounced than in conventional ones.
The ability of highly deformable materials to undergo very large
deformations before failure is mainly due to their noticeable rearrangement of the molecular network with a significant decrease of the internal entropic state. Neglecting such an entropic effect can lead to erroneous underestimations of the safety level against defect-driven failure. In the present research, the mechanics of highly deformable materials is discussed by examining silicone-based notched and cracked plates. Experimental, numerical and theoretical aspects of the involved phenomena are analyzed in order to provide an explanation of the mechanism of defect resistance in such a class of materials from a physics-based point-of-view.
Keywords: Highly deformable materials; Flawed structures, Polymer, Defect tolerance.
* Corresponding author. Tel.: +39 0521 905910; fax: +39 0521 905924. E-mail address: [email protected]
Nomenclature
2a, 2d
Principal axes of the elliptical notch (maximum and minimum, respectively)
b
Average length of the chain segments in the polymeric chain
b
Body force vector
c
Concentration of the cross-links in the polymer per unit volume
E
Young modulus of the material
Eij
Green-Lagrange strain tensor components
f (a / W )
Function of the ratio of notch to plate size
F , Fij xi / X j
Macroscopic deformation gradient tensor
F
Macroscopic deformation rate tensor
G
Energy release rate at fracture
G 4 2
Fracture energy of a penny shaped crack with radius
k
Boltzman’s constant
K t f (a / W ) K t Stress concentration factor for a finite notched plate
Kt
Stress concentration factor for an infinite notched plate
n, N
Number of linked chains in the unit volume of material and number of chains’ segments, respectively
P
First Piola-Kirchhoff stress tensor
r0 , r
End-to-end vector of a chain in the initial and in a generic state, respectively
t
Surface traction force vector
T
Absolute temperature
U (r)
Energy stored in a single chain having an end-to-end vector r
W
Potential energy of the external loads
y ,0
Remote tensile strain in the y direction
y
Strain rate acting in the y direction
(r, t )
Total variation of the chains’ end-to-end vector r distribution function at
2
the time t
t (r, t )
Distribution of the chains’ end-to-end vector r density at a generic time t
0 (r)
Distribution of the chains’ end-to-end vector r density in the initial stressfree state
(r, t ) t (r, t ) 0 (r )
Distribution difference of the chains’ end-to-end vector r density in the
on (r, t ) , off (r, t )
Positive and negative contributions rate to the stretch distribution function
current and in the stress-free state
rates, respectively 0
Density of attached cross-links at equilibrium
Fracture energy per unit surface
λ
Macroscopic stretch of the polymer
S
Surface stretch
Poisson’s coefficient of the material Average radius of the voids initially present in the polymer
Curvature radius at the notch root
σ
Cauchy stress tensor
V , V
Strain energy density (unit volume) contained in a stretched elastomer and its increment, respectively
3
1. Introduction Many structural components containing defects such as holes, notches, cracks, geometric discontinuities, etc. are observed to fail for a stress level lower than that for the corresponding unflawed components [1-3]. Defect-driven failure usually obeys different physical laws than those for classical strength-based failure in unflawed components. In the present study, the influence of defects of different severity contained in structures made of materials which withstand high deformations (generally named soft materials) is analyzed. Such materials are for instance: liquid crystal elastomers, colloids, polymers, foams, gels, granular materials, and many biological materials. The polymers herein examined belong to such a class of materials, and are characterized by a fully amorphous microstructure being composed – at the molecular level – by very long linear entangled macromolecules in polymer melts or polymer solutions joined together in several points called cross-links (shortly indicated by XL, Fig. 1a). The structure of such materials consists of a three-dimensional network of polymer chains closely linked, and their mechanical response is mainly affected by the amount of cross-links in the unit volume. The deformation of such a class of materials is characterized by the relative motion of the long entangled macromolecules (also known as ‘reptation’, a term coined by the physics Nobel laureate Pierre-Gilles de Gennes, the founder of the science of soft matters physics [4, 5]). Relevant results in the physics and chemistry of macromolecules-based materials, such as the polymeric ones, have been achieved by another physics Nobel prize winner, Paul J. Flory [6], with his fundamental theoretical and experimental research work. A remarkable difference between the mechanical behaviour of a traditional material and that of a highly deformable polymeric one is the different resisting mechanism underneath the observable macroscopic response to mechanical actions. The bonding strength and stiffness 4
existing between atoms entail the response of traditional materials, while the entropic-related effects, typical of elastomers, are mainly responsible for the behaviour of very deformable amorphous materials, since their polymeric chains (connected to each other through covalent bonding in several cross-link junctions, Fig. 1b) can be easily deformed by their geometrical rearrangement. The reduction of the level of high disorder in the entangled polymeric chain network under a mechanical stress increases the material’s internal energy and reduces the corresponding entropic level, so that the two above internal variables (deformation energy and entropy) can be related each other (Fig. 1c).
In summary, the microstructural nature of
polymeric materials entails their physical properties to be mainly governed by the degree of cross-linking existing between the constituting chains.
Please insert here Fig. 1
In the present paper, the problem of defect-induced failure in elastomeric sheets is examined from both experimental and theoretical point-of-view. The influence of the degree of severity of the initial defect and the strain rate effect are both analyzed: in particular, the strain rate influence on the mechanical response is examined to account for the creep phenomenon that has a relevant role in such a class of materials, and can significantly affect their defect-tolerance ability. Experimental tests involving various degrees of severity of the initial defect under different strain rates are interpreted through a theoretical model based on a statistical description of the material at the micromechanical level.
5
2. Some basic concepts of the mechanics of polymers 2.1. Statistics-based description of mechanics of polymers The mechanical response of a polymer network, such as that characterizing an elastomeric material, can be obtained by analyzing the elasticity behaviour of a single polymer chain. A chain is the part of the linear polymeric macromolecule placed between neighboring cross-links (Fig. 1b, c). According to the Kuhn’s theory of rubber elasticity [7-12], typically formulated on the basis of the random walk theory (Fig. 2), the entropy variation S between two states (say reference state 0 and final state 1 , Fig. 1b, c) in a given small volume of the material chain is given by [7, 9]: 1 S k ln( P1 / P0 ) , with Pi (r ) 2 2N
3/ 2
r N r exp 2 N 2
2
(1)
where k 1.38 1023 J/K is the Boltzmann constant, whereas N , r 0, 2 b 2 / 3 are the number of segments in the polymer chain, the average value of the end-to-end distance (with sign) of one chain, and the variance of the chain length (being b the average length of one segment of the chain, Fig. 2(a1)), respectively. The distribution function P i (r ) provides the probability that the end-to-end vector of one single chain is equal to r (Fig. 2b).
Please insert here Fig. 2
6
For a single stretched polymer chain, the variation of the deformation work (F) produced by the stretch acting on the material (quantified by the deformation tensor F ), can be related to its changing of entropy as follows [7]:
(λ ) T S ½ kT 2x 2y 2z 3
(F) ½ kT tr(FT F) 3 ,
(2)
where (λ ) represents the variation of the deformation work (F) in the particular case of a material stretched along only the three coordinate axes by the stretches x , y , z . The energy per single chain given by Eq. (2a) can be used to determine the energy stored per unit volume of material by adding up the contribution of all the single chains contained in such a volume. Once the chain end-to-end distance distribution in the material is known, the overall energy can be evaluated. The chain end-to-end distance distribution in polymers is usually assumed to be described by a Gaussian function, (r, t ) :
3 (r, t ) 2 2Nb
3/ 2
3 r2 c exp c r 2 exp 2 2 Nb 3/ 2
(3)
where the term c represents the cross-links concentration per unit volume. On the other hand, the variation of the deformation energy stored in one single chain when its initial end-to-end distance r0 becomes equal to r Fr0 is given by
U (r )
3kT 2 Nb
2
r2
3kT 2 Nb
2
(Fr ) 0
2
r02
(4)
Finally, by considering Eqs (3) and (4), the variation of the deformation energy stored in the unit volume of polymer can be expressed as follows:
7
V (r, t ) (r, t )U (r) dr
(5)
For a stretched volume of material, whose deformation is quantified by the deformation gradient F , equation (5) can explicitly be written as follows: V (F)
3nkT 2 Nb 2
0 dN (r, N ) (Fr0 )
2
r02 dr
3nkT tr(FT F) 3 (r )dr dr 2 2 Nb Nb 2 / 3
(6)
nkT AkT tr(FT F) 3 tr(FT F) 3 2 2
where it is assumed, as is typical within the rubber elasticity theory, that the deformation process affects only the energy U (r ) per single chain, while it does not affect the distribution function
(r, t ) .
2.2. Interpretation of the effect of deformation on the stored energy The energy variation produced by the deformation occurring in the polymeric network has been written in Eq. (5) by using the energy variation per unit chain given by Eq. (4). On the other hand, the effect of stretching applied to the network’s chains (Fig. 2(a2)) can also be interpreted as a modification of the end-to-end length distribution (r, t ) of the entangled linear polymeric molecules (Fig. 2b). As a matter of fact, due to chains extension, the concentration of links for a given end-to-end distance is not equal to that in the stress-free state [7]. However, the total number of cross-links per unit volume of material does not change because of the stretch (at least if no failure occurs) and, therefore, the new probability distribution function must preserve its area regardless of the amount of the applied stretch (Fig. 2b). The variation of the internal energy according to such an interpretation leads to:
8
V (r, t ) V (r, t ) V (r,0)
t (r, t ) 0 (r)U (r) dr
(7)
In general, the infinitesimal variation of the probability distribution function can be expressed as follows [13]:
dt (r, t ) dt (r, t ) XL dt (r, t ) F t (r, t ) XL t (r, t ) F dt
(8)
where the first term dt (r, t ) XL represents the variation of the distribution function for constant cross-links (i.e. no cross-links are lost or created in the time interval dt ), and t (r, t ) F is the variation of the distribution function due to the variation of the cross-link density but at constant deformation gradient F . In the case of constant cross-links, the variation dt (r, t ) XL can explicitly be expressed by performing a power series expansion of the probability distribution function with respect to the end-to-end distance r , i.e. we have to expand the quantity (r r) . After some calculations, we get [13]:
t (r, t ) XL
t (r) 1 F F r t (r) tr F F 1 r
(9)
On the other hand, if we want to account for the changing of the cross-links in the polymer network, the term t (r, t ) F must be evaluated. The increment of the distribution function
t (r, t ) F for a constant deformation gradient F can also be exploited to simulate the creep phenomenon. It is worth mentioning that the cross-links concentration in the unit volume can be considered
to correspond to the total number of linked chains in the network, i.e. c(t ) (r, t )dr . In the
9
case of no variation of the cross-links numbers per unit volume, such a concentration must be constant during the deformation process, i.e. c(t ) c0 const., t . The above presented formulation for the mechanics of polymeric materials – based on the micro mechanics of the stretched elastomer chains and on the statistical distribution of their endto-end vector – has the advantage of allowing to easily keep into account for large strain features and to consider the possibility for cross-links internal rearrangement (through the term t (r, t ) F in Eq. (8)) that can be exploited to model viscoelastic phenomena; however this last issue will be not addressed in the present study.
2.3. Failure model for polymer chains network Failure of soft elastomers has also been explained through the growth of spherical-like cavities at the mesoscale level, due of the very low elastic modulus compared to the cohesive strength: these mechanical properties are responsible for the (tensile) pressure sensitivity of this class of materials [14-16]. Studies concerning the failure mechanisms in elastomers revealed that microscopic voids typically grow and coalesce into macroscopic ones under hydrostatic tension; sometimes such a process in rubber-like materials is also termed as cavitation [17]. The process of cavity growth and expansion usually exhibits a stiffening behaviour due to unfolding of the long constituents molecules of the polymer [14-17], and other studies have concluded that failure in polymers is driven by fibril creep mechanism [18]. This entails a stable damage phenomenon and justifies, from a qualitative point of view, the defect tolerance in soft polymeric materials. Since the elastomeric materials behave elastically with a good approximation (at least for sufficiently high
10
strain rates), the Griffith’s energy balance for fracture failure referred to a penny shaped circular crack of radius can be written as follows: W G / 0
(10)
where W , , G 4 2 are the potential energy of the external force, the deformation energy stored in the material and the fracture energy, respectively ( being the fracture energy per unit surface). Since the external load is applied statically and the fracture failure is an instantaneous phenomenon, the force potential is zero ( W 0 ) and, therefore, the above energy balance leads to the well-known fracture condition:
/ 4 2
(11)
where the decreasing nature of the internal energy of the material during the fracture process is taken into account. The relationship between the fracture energy and the critical tensile pressure, that produces an unstable expansion of an existing spherical cavity with initial radius in an infinite elastic neo-Hookean medium, can be determined by writing the energy release rate G evaluated between two penny-shaped cracks with radius and + d , respectively [16]: ( ) ( d ) d 0 2d
G lim
(12)
By assuming that the crack can be expanded in a spherical void with the same radius without any further energy, the term ( ) can be evaluated through the solution of the elastic problem related to the case of a spherical cavity in an infinite elastic medium inflated by a pressure p [19]:
11
1 2S 2S1 G 2 E 3
(13)
where the function ( ) has been expressed by making use of the surface stretch S and the Young modulus of the material. The above surface stretch S can be also related to the pressure p acting inside the spherical void through the relation p(S ) E (5 4S1 S4 ) / 6 , with
S S ' / S 1 2d 1 , S , S ' being the external surfaces of the initial and final spherical void, respectively. Equation (13) can be expressed only in terms of the pressure p by substituting the relation p(S ) .
For sufficiently large pressure values, namely p 11E / 18 [16], Eq. (13)
becomes: G
2 E 16 / 25 3 3p 2 3 [1 (6 p / 5E )] 2 E
(14)
By assuming that the energy release rate G is equal to the fracture energy of the material,
G , the critical pressure pc leading to failure can be determined: 2 E 16 / 25 3 3 pc 0, 3 [1 (6 pc / 5E )]2 2 E
with p 5E / 6
(15)
The above pressure corresponds to the hydrostatic stress in the material, i.e. p ii / 3 , and thus the failure criterion for void growth can be stated on the basis of the stress components as follows: 2 E 16 / 25 3 ii 0 3 [1 (2 ii / 5E )]2 2 E
12
(16)
That requires the knowledge of the average radius of the voids which are initially in the material. As soon as the critical condition for void expansion is attained, it can be assumed that the stress state in that point vanishes because of the material failure occurring in the small region around such a point of cavitation.
2.4. Stress state evaluation According to the statistics-based theory of elasticity typically adopted for rubber materials, the knowledge of the chains’ length distribution allows for the evaluation of the internal energy variation with respect to the initial reference state, V (r, t ) . The first Piola stress tensor can be obtained as the gradient of the internal energy variation with respect to the components of the deformation gradient, i.e. P V (r, t ) / F . By using Eqs (7) and (11), the above derivation provides the expression of the first Piola stress tensor and that of the Cauchy stress tensor at a given time t of the deformation process:
P ( F, t )
V (r, t ) F 1r (r, t )F T U (r) dr F r
σ P(F, t ) FT / J
(17a)
(17b)
J det F being the determinant of the deformation gradient. Further, the Green’s strain tensor
can be evaluated through the well-known relationship ε ½(FT F I) [20].
13
3. Numerical implementation of the statistics-based model for polymers failure The above described micromechanics-based model can easily be implemented in a computational tool such as in a model based on the finite element method. The non-linearity of the mechanical response of the material, both in the cases involving no failure – where the material shows a hyperelastic response according to the adopted statistical approach for rubber elasticity – and in the case of chains failure, requires to solve incrementally the problem. Referring to a problem discretized through finite elements, once the internal energy is known for a given applied load or displacement, the first Piola stress tensor P and the Cauchy stress tensor σ can be evaluated at each finite element’s Gauss point through Eqs (17a, b), and used to evaluate the vector Φ e of the unbalanced nodal forces of each finite element :
Φe
B
Ωe
T
σ N b dΩ
N
T
t d Pe 0
e t
(18a)
to be employed in the iterative solution process after assembling the unbalanced nodal force values of each element into the corresponding global vector related to the whole structural component being analyzed. In Eq.(18), e , e t are the current domain and traction boundary configurations of the finite element in turn, B, N, b, t are the compatibility matrix, the matrix of the shape function, the vector of the volume and the vector of the traction forces applied to the finite element’s boundary at the current time instant, respectively. Further, Pe is the nodal vector of the external forces applied to the finite element e . The initial cross-link concentration of the material can be obtained by the well-known relationship c(0)
(r,0)dr E /(3 k T ) that provides a connection between the standard
Young modulus E and the junction density per unit volume of the polymeric network [7, 11]. 14
At every time instant of the loading process, the internal energy V (r, t ) (assessed with respect to the stress-free state through Eq. (7)) is computed after updating the distribution function by means of the expression t (r, t dt ) t (r, t ) dt (r, t ) dt , where the increment dt (r, t ) is evaluated by using Eq. (8). In other words, the Cauchy stress tensor is determined
by the following constitutive relationship (r, t ) σ F 1r (r, t )F T U (r ) dr FT J 1 r
(18b)
that relates σ to the deformation gradient tensor F through some evolving material-related parameters (embedded in the actual distribution function (r, t ) ) and to the energy stored in a single chain given by the standard expression U (r) (see Eq. (4)) arising from the Gaussian statistics of the polymer chains network [7]. The above proposed micromechanics-based formulation of the failure phenomenon does not allow to represent the formation of crack-like ruptures in the material, but it considers the damage as a continuous field variable through the relaxed stress field produced by the chains failure. No discontinuity is allowed to occur and, therefore, the material maintains its continuity. Fully nonlinear geometrical analyses are carried out and, therefore, the integrals (18) are evaluated in the current deformed configuration of the elements in order to account for deformation in the equilibrium states evaluated along the loading path.
4. Experimental tests In order to quantify the mechanical response of flawed elastomeric structural components under tension up to the final failure, different notched thin plates are examined. Pre-notched sheets
15
made of a common silicone polymer obtained from the cross-linking of two vinyl-terminated polydimethylsiloxane matrix and an hydride terminated siloxane curing agent, with a Pdcatalyzed hydrosilylation, also commercially known as Sylgard®, are tested under tension at different strain rates. Then, experimental results are elaborated with Digital Image Correlation (DIC) technique to obtain the deformation field in the samples during the tests. The tested elastomeric sheets have initial elastic modulus equal to about E 0.98MPa and Poisson’s ratio 0.42 , whereas the geometric characteristics of the specimens are shown in Tab.1. In particular, three groups of notched plates have different values of the root radius ( 1.25 mm , 0.312 mm and 0.005 mm ), whereas the last one is a cracked specimen (corresponding to the asymptotic case of a notch with zero curvature radius). In Tab. 1, the geometrical notch parameters a, d , and the corresponding stress concentration factor Kt ( ) for each actual finite plate are reported: such a local stress field parameter is defined as the ratio
Kt max / nom between the maximum to the nominal stresses evaluated at the notch root [21, 22]. In the case of a plate with an elliptical notch with semi-axes a, d (Tab. 1), the minimum curvature at the root of such an elliptical notch profile and the corresponding stress concentration factor in the case of an infinite plate can be expressed as follows:
d2 / a,
K t 1 2a / d 1 2 a /
(19)
In the case of a finite plate with the same notch, the corresponding stress concentration factor can be obtained as Kt f (a / W ) K t , where f (a / W ) is a function of the ratio of notch size to plate size [23]. The use of different values of notch radius enables to understand the effect of the peak stress as well as of the stress gradient on the mechanical response and failure mechanism of highly deformable polymeric materials: as a matter of fact, the local reorganization and the 16
chains re-arrangement in the most stressed region of the body is expected to enable a noticeable defect tolerance capability of the component that should tend to be nearly defect-severity insensitive. The specimens have been tested by imposing an increasing displacement (t ) / 2 , applied with a constant strain rate in the y-direction at each horizontal edge of the plate (see right-hand side of Tab. 1) up to the final failure. Three different strain rates have been applied to the specimens: (i) y1 3.846 103 s 1 , (ii) y 2 9.615 104 s 1 and (iii) y 3 1.603 104 s 1 . These deformation velocities correspond to apply a displacement increment to the silicone sheet equal to = 1mm/2.5s, 1mm/10s, 1mm/60s, respectively. The use of different strain rates enables to examine the effect of relaxation that could eventually occur in such materials: slower strain should allow a better rearrangement of the polymeric network in highly stressed regions, enabling a better defect insensitivity behaviour.
Please insert here Tab. 1
In Figs 3 to 5, the deformed shapes of the specimens for different applied nominal remote strain levels ( y ,0 (t ) (t ) / 2L ) are shown for the three strain rates y1, y 2 , y 3 , applied in the tests. Irrespective of the initial notch severity, the maximum applicable remote strain before failure can be noticed to not depend on the stress concentration factor of the notch. Because of the notch blunting and the local instability occurring close to the notch edges [23], the notch effect results to be similar in all cases, and the initial notch root value does not heavily affect the mechanical behaviour of the sheet. Moreover, note that slower strain rates allow, in general, to get a bigger maximum strain value before failure. 17
Please insert here Fig. 3
Please insert here Fig. 4
Please insert here Fig. 5
Differences in the maximum strain values at failure can be also explained by accounting for the internal micro defects in the polymer. As a matter of fact, the production of the elastomer entails the inclusion of micro air bubbles that have to be removed by an autoclave process. However, such a procedure cannot completely remove such inclusions, and the presence of small bubbles is unavoidable. Note that their detrimental effect is more severe in specimens with blunt notches than in those with sharp notches: in fact, the high stress levels produced by defects with small values of the tip curvature radius can lead to cavitation failure (see Eq. (16)) of spherical inclusions with smaller values of radius . In order to study the effect of the strain rate on the mechanical response of the material, several relaxation tests have been performed. In Fig. 6, the stress-time relaxation tests are displayed when a nominal strain equal to y,0 40% is applied to the material. As can be noted, the strain rate has some slight effect on the stress against time relationships due to the polymer chains internal rearrangement with time. In other words, the chains tend to align more and more along the loading direction while the applied strain is kept on the material. This stress relaxation and chains reordering contribute to reach higher strain values before failure of the material.
18
Please insert here Fig. 6
Failure surprisingly takes place at a level of the remote nominal strain y ,0 that does not reduce with the notch severity, i.e. by increasing the SCF value K t ( ) .
The elevated
deformation of the material allows the stress value in the vicinity of the notch root to remain limited, and the failure condition results to be almost independent of the initial notch radius; further, the complete rearrangement of the notch shape is also responsible for the direction of the crack initiation not to be in pure Mode I (Figs 3 to 5). On the basis of the strain energy at failure (Fig. 7), the fracture energy of the material has been estimated to be equal to about 2.5 104 N / m , whereas the voids radius has been estimated to be approximately equal to 10 4 m . These values are used in the numerical simulations shown in the following section for predicting the cavitation-like failure in the material.
Please insert here Fig. 7
5. Numerical analysis and discussion The theoretical mechanical model for elastomers presented in Sub-Sections 2.2, 2.3 and 2.4 can conveniently be implemented in a FE code by evaluating the stress state through the developed micromechanical approach applied at the Gauss point level. In the present research, the above model has been implemented in a non-linear 2D FE code, and the simulation of the mechanical response of notched elastomeric sheets has been compared with the experimental outcomes. 19
In Fig. 8, the load-displacement curves obtained numerically and from the experimental tests are displayed up to a maximum applied strain equal to about 33%. The stiffness of the elements is correctly predicted by the FE model despite some irregularities in the numerical results (such irregularities are due to the iterative procedure employed).
Please insert here Fig. 8
Figure 9 displays the deformed patterns provided by the FE analyses for plates No.1a and No.4a, for two levels of the remote applied strain y ,0 . The evolution of the deformed shape and the development of the local failure at the notch tip is also provided by the numerical model, where the mechanical response of the polymer has been implemented according to Eq. (17a) and the cavitation-like failure criterion is described by Eq. (16).
Please insert here Fig. 9
Finally, the failure patterns developed during the experimental tests for the specimens No. 1a and 4a are displayed in Fig. 10. As can be observed, the failure modes resemble those predicted by the numerical analyses, that is to say, a nearly dovetail-like failure shape is recognizable.
Please insert here Fig. 10
According to the obtained results, the examined soft polymer shows the ability to withstand macroscopic defects at high applied strain levels, irrespective of the local curvature radius of the considered notch. This unexpected behaviour comes from noticeable local reshape of the defect 20
and from the mechanism of entanglement reduction of the polymeric chains during the stretch process. Moreover, small defects embedded in the polymer, like micro voids, are responsible for the progressive damage in the material which fails due to the expansion and coalescence of the existing micro-bubbles under the tensile hydrostatic stress state. The complex mechanical response of polymers, characterized by anisotropy, time dependent and uncommon damage and fracture behaviour, has been analyzed in Ref. [24] for monotonic and cyclic loading by adopting different material strain-energy density functions (corresponding to a simple hyperelastic model), whereas the effect of strain rate and temperature influence in semicrystalline thermoplastic polymer has been examined in Ref. [25].
It has been
experimentally shown that increasing the temperature or lowering the strain rate entail a softer response of the polymer due to the internal cross-links rearrangement of the material. Such a phenomenon can be quantitatively described by the present model through the term t (r, t ) F in Eq. (8). As far as the damage mechanism is concerned, the above observed behaviour is similar to that of a hierarchical material in which the macroscopic failure proceeds by involving (micro) failures at lower scales as has been shown to occur in several natural materials [26] that can be considered to be optimized with respect to the defect tolerance capability.
6. Conclusions The defect tolerance against failure in highly deformable materials has been examined in the present paper. A theoretical model based on the so-called rubber elasticity has been developed to explain the ability of such a class of material to withstand very high strain values, irrespective of the notch or defect severity.
21
Notched and cracked elastomeric plates made of a silicone polymer have been tested by applying an increasing strain (three different values of strain rate have been analyzed) up to the final failure. The tests have shown that the failure condition is almost independent of the initial stress concentration factor, because the high deformation of the elastomeric sheet around the flawed region smoothes out the peak stress even for an initially sharp straight crack. Then the observed mechanical behaviour is interpreted through a model based on the micromechanics of failure in elastomeric-like materials composed by long entangled crosslinked chains: the proposed approach can macroscopically describe the failure of elastomeric materials on the basis of the micro-damage mechanisms occurring at high strain levels in crosslinked polymeric chains. Strain rate effect has been found to have some influence in the failure behaviour of highly deformable flawed polymer specimens, but it has been neglected in the present theoretical model since we focused mainly on the development of the deformation model and failure for short-time loading for such a class of materials. Further studies and enrichments of the developed theoretical framework are needed. Mechanical failure of soft materials, that are broadly used in many modern applications such as in bioengineering dealing with biological tissues, gels, liquid crystals, etc., can conveniently be examined through the proposed model, which is suitable to be implemented in computational codes.
Acknowledgements The Authors gratefully acknowledge Prof. E. Dalcanale of the Dept. of Chemistry of Parma Univ. for the preparation of the Sylgard® silicone polymer sheet specimens.
22
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M. Doi, Soft Matter Physics, Oxford Univ. Press, UK, 2013.
[8]
W. Kuhn, Dependence of the average transversal on the longitudinal dimensions of statistical coils formed by chain molecules, J. Polymer Sci. 1 (1946) 380.
[9]
P.J. Flory, Statistical Mechanics of Chain Molecules, Hanser-Gardner, Cincinnati, Ohio, 1989.
[10] L.R.G. Treloar, Physics of Rubber Elasticity, Oxford University Press, 1975. [11] L.R.G. Treloar, The elasticity and related properties of rubbers, Rep. Prog. Phys. 36 (1973) 755–826. [12] E. Guth, H.M. James, Elastic and thermoelastic properties of rubber like materials, Ind. Eng. Chem. 33(5) (1941) 624–629. [13] R. Brighenti, F.J. Vernerey, A simple statistical approach to model the time-dependent response of polymers with reversible cross-links, Composite Part B: Eng. (2016), in press (DOI: 10.1016/j.compositesb.2016.09.090). [14] Y.Y. Lin, C.Y. Hui, Cavity growth from crack-like defects in soft materials, J. Fract. 126(3) (2004) 205–221. [15] C. Fond, Cavitation criterion for rubber materials: a review of void-growth models, J Pol. Sci.: Part B: Polymer Physics 39 (2001) 2081–2096.
23
[16] Y.Y. Lin, C.Y. Hui, Cavity growth from crack-like defects in soft materials, J. Fract. 126 (2004) 205–221. [17] Y. Lev, K.Y. Volokh, On cavitation in rubberlike materials, J. App. Mech. 83(4) (2016) 0044501-1–0044501-4. [18] M. Jie, C.Y. Tang, Y.P. Li, C.C. Li, Damage evolution and energy dissipation of polymers with crazes, Theor. Appl. Fract. Mech. 28(3) (1998) 165–174. [19] A.N. Gent, C. Wang, Fracture mechanics and cavitation in rubber-like solids, J. Mat. Sci. 26(12) (1991) 3392–3395. [20] G.A. Holzapfel, Nonlinear solid mechanics: a continuum approach for engineering, John Wiley & Sons, West Sussex, England, 2000. [21] W. C. Young, R. G. Budynas, Roark’s formulas for stress and strain, McGraw-Hill, 7th edition, 2002. [22] W. D. Pilkey, D. F. Pilkey, Peterson's Stress Concentration Factors, 3rd Edition, Wiley, 2008. [23] R. Brighenti, A. Spagnoli, A. Carpinteri, F. Artoni, Notch effect in highly deformable material sheets, Thin-Walled Structures 105 (2016) 90–100. [24] D.A. Serban, L. Marsavina, V. Silberschmidt, Behaviour of semi-crystalline thermoplastic polymers: Experimental studies and simulations, Comput. Mat. Sci. 52 (2012) 139–146. [25] D.A. Serban, G. Weber, L. Marsavina, V.V. Silberschmidt, W. Hufenbach, Tensile properties of semi-crystalline thermoplastic polymers: Effects of temperature and strain rates, Polymer Testing 32 (2013) 413–425. [26] R. Mirzaeifar, L.S. Dimas, Z. Qin, M.J. Buehler, Defect-tolerant bioinspired hierarchical composites: simulation and experiment, ACS Biomater. Sci. Eng. 1 (2015) 295−304.
24
Figures and captions
(a)
0
1
(b)
(c)
Fig. 1. (a) Scheme of a 3D polymer network. Simple sketch of : (b) unstretched configuration 0 and (c) stretched configuration 1 of the cross-linked polymeric chains.
25
Chains microscopic end to end distance density probability
b
z
b r0
(a1)
y
O
x
F
z
r
(a2)
F=
[
(b)
(0)
(,t) 1 (1)
y
O
]
0()
0
stretch ( )
Chains end-to-end distance, r =
x
Fig. 2. (a1) Scheme of an undeformed polymer chain and definition of its end-to-end vector; (a2) corresponding deformed chain; (b) probability distribution functions of the chains lengths before and after deformation.
Spec. No.
1a
W
2a
(mm) (mm) 114 40
2d
t
(mm) (mm) 10 2.10
2a/W
a/d
(---) (---) 0.351 4
Kt ( )
(mm) (---) 1.250 12.43
y (s-1)
y1
1b
114
40
10
1.50
0.351
4
1.250 12.43
y 2
1c
114
40
10
1.95
0.351
4
1.250 12.43
y 3
2a
114
40
5
1.50
0.351
8
0.312 24.67
y1
2b
114
40
5
2.25
0.351
8
0.312 24.67
y 2
2c
114
40
5
2.25
0.351
8
0.312 24.67
y 3
3a
114
40
2
1.60
0.351
20
0.005 61.73
y1
3b
114
40
2
2.00
0.351
20
0.005 61.73
y 2
3c
114
40
2
1.80
0.351
20
0.005 61.73
y 3
4a
114
40
<0.05
2.15
0.351 >400
~0 26
---
y1
4b
114
40
<0.05
1.80
0.351 >400
~0
---
y 2
4c
114
40
<0.05
2.00
0.351 >400
~0
---
y 3
Tab. 1. Geometric characteristics of the tested specimens and related stress concentration factors K t ( ) . Specimens labeled as na, nb and nc (with n=1, 2, 3, 4) correspond to the strain rates y1, y 2 , y 3 , respectively.
27
Fig. 3. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y1 3.846 103 s 1 strain rate.
28
Fig. 4. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y 2 9.615 104 s 1 strain rate.
29
Fig. 5. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y3 1.603 104 s 1 strain rate.
30
3.6E+005
2 nd cycle
y,0= 0.4
0
(b)
3 rd cycle
2.8E+005 100
3rd cycle
1 st cycle 3.0E+005
1st cycle
3.2E+005
2nd cycle
1
factor, f (t)
Stress, x (Pa)
(a) 3.4E+005
200
300
400
500
0
600
600
1200
1800
2400
3000
Time, t (s)
Time, t (s)
Fig. 6. (a) Stress relaxation test under an applied nominal strain equal to y,0 40% . (b) Three cycles are examined, namely y (t ) f (t ) y,0 .
(a1)
(a2) Force, F (N)
failure
spec. 1a spec. 2a
200 150 100 50
(b) 0 0
10
20
30
40
Displacement, (mm)
50
60
Fig. 7. Crack paths at failure in the specimen 1a (a1) and 2a (a2); (b) load-displacement curve for the same specimens up to the final fracture collapse.
31
Force, F (N)
Applied force, F (N)
(a)
40
spec. No. 1a 30 20 exp. results FE results
10
(b)
40
spec. No. 4a 30 20 exp. results FE results
10 0
0 0
5
10
15
Applied displacement, (mm)
20
0
5
10
15
Applied displacement, (mm)
20
Fig. 8. Experimental and numerical force F against applied displacement (see Tab. 1) for: (a) the notched specimen 1a ( 2b 10mm ), and (b) the cracked specimen.
(a1)
(b1)
(a2)
(b2)
Fig. 9. Deformed patterns and Green’s strain Eyy determined by the model of failure for elastomers implemented in a FE code. Specimens No. 1a (a1, a2) and No. 4a (b1, b2), for a remote nominal applied strain y,0 18% (a1, b1) and y,0 33% (a2, b2). Details of the local failure patterns are also reported in the two magnifications.
32
(a)
(b)
Fig. 10. Detail of the failure pattern at the notch tip for specimens 1a (a) and 4a (b), developed during the experimental tests.
33
Highlights
- Failure defect sensitivity in polymers is examined - A theoretical mechanical model is formulated based on entropy concept - A theoretical failure mechanism of polymers is taken into account - Failure experimental results of notched sheets are presented - Experimental results are compared with numerical outcomes
34
S0167-8442(16)30312-3 http://dx.doi.org/10.1016/j.tafmec.2016.12.005 TAFMEC 1794
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
11 October 2016 12 December 2016 27 December 2016
Please cite this article as: R. Brighenti, A. Carpinteri, F. Artoni, Defect sensitivity to failure of highly deformable polymeric materials, Theoretical and Applied Fracture Mechanics (2016), doi: http://dx.doi.org/10.1016/j.tafmec. 2016.12.005
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
DEFECT SENSITIVITY TO FAILURE OF HIGHLY DEFORMABLE POLYMERIC MATERIALS
Roberto Brighentia*, Andrea Carpinteria, Federico Artonia a
Dept. of Civil-environmental Engineering & Architecture, Univ. of Parma, Viale delle Scienze 181A, 43124 Parma, ITALY
Abstract The capability of materials to bear loads even in presence of defects like cracks, notches or generic geometric discontinuities is usually indicated as flaw tolerance, and is crucial in modern safety design of structural components. Such a tolerance capability can be remarkable in highly deformable materials (also called soft materials), usually much more pronounced than in conventional ones.
The ability of highly deformable materials to undergo very large
deformations before failure is mainly due to their noticeable rearrangement of the molecular network with a significant decrease of the internal entropic state. Neglecting such an entropic effect can lead to erroneous underestimations of the safety level against defect-driven failure. In the present research, the mechanics of highly deformable materials is discussed by examining silicone-based notched and cracked plates. Experimental, numerical and theoretical aspects of the involved phenomena are analyzed in order to provide an explanation of the mechanism of defect resistance in such a class of materials from a physics-based point-of-view.
Keywords: Highly deformable materials; Flawed structures, Polymer, Defect tolerance.
* Corresponding author. Tel.: +39 0521 905910; fax: +39 0521 905924. E-mail address: [email protected]
Nomenclature
2a, 2d
Principal axes of the elliptical notch (maximum and minimum, respectively)
b
Average length of the chain segments in the polymeric chain
b
Body force vector
c
Concentration of the cross-links in the polymer per unit volume
E
Young modulus of the material
Eij
Green-Lagrange strain tensor components
f (a / W )
Function of the ratio of notch to plate size
F , Fij xi / X j
Macroscopic deformation gradient tensor
F
Macroscopic deformation rate tensor
G
Energy release rate at fracture
G 4 2
Fracture energy of a penny shaped crack with radius
k
Boltzman’s constant
K t f (a / W ) K t Stress concentration factor for a finite notched plate
Kt
Stress concentration factor for an infinite notched plate
n, N
Number of linked chains in the unit volume of material and number of chains’ segments, respectively
P
First Piola-Kirchhoff stress tensor
r0 , r
End-to-end vector of a chain in the initial and in a generic state, respectively
t
Surface traction force vector
T
Absolute temperature
U (r)
Energy stored in a single chain having an end-to-end vector r
W
Potential energy of the external loads
y ,0
Remote tensile strain in the y direction
y
Strain rate acting in the y direction
(r, t )
Total variation of the chains’ end-to-end vector r distribution function at
2
the time t
t (r, t )
Distribution of the chains’ end-to-end vector r density at a generic time t
0 (r)
Distribution of the chains’ end-to-end vector r density in the initial stressfree state
(r, t ) t (r, t ) 0 (r )
Distribution difference of the chains’ end-to-end vector r density in the
on (r, t ) , off (r, t )
Positive and negative contributions rate to the stretch distribution function
current and in the stress-free state
rates, respectively 0
Density of attached cross-links at equilibrium
Fracture energy per unit surface
λ
Macroscopic stretch of the polymer
S
Surface stretch
Poisson’s coefficient of the material Average radius of the voids initially present in the polymer
Curvature radius at the notch root
σ
Cauchy stress tensor
V , V
Strain energy density (unit volume) contained in a stretched elastomer and its increment, respectively
3
1. Introduction Many structural components containing defects such as holes, notches, cracks, geometric discontinuities, etc. are observed to fail for a stress level lower than that for the corresponding unflawed components [1-3]. Defect-driven failure usually obeys different physical laws than those for classical strength-based failure in unflawed components. In the present study, the influence of defects of different severity contained in structures made of materials which withstand high deformations (generally named soft materials) is analyzed. Such materials are for instance: liquid crystal elastomers, colloids, polymers, foams, gels, granular materials, and many biological materials. The polymers herein examined belong to such a class of materials, and are characterized by a fully amorphous microstructure being composed – at the molecular level – by very long linear entangled macromolecules in polymer melts or polymer solutions joined together in several points called cross-links (shortly indicated by XL, Fig. 1a). The structure of such materials consists of a three-dimensional network of polymer chains closely linked, and their mechanical response is mainly affected by the amount of cross-links in the unit volume. The deformation of such a class of materials is characterized by the relative motion of the long entangled macromolecules (also known as ‘reptation’, a term coined by the physics Nobel laureate Pierre-Gilles de Gennes, the founder of the science of soft matters physics [4, 5]). Relevant results in the physics and chemistry of macromolecules-based materials, such as the polymeric ones, have been achieved by another physics Nobel prize winner, Paul J. Flory [6], with his fundamental theoretical and experimental research work. A remarkable difference between the mechanical behaviour of a traditional material and that of a highly deformable polymeric one is the different resisting mechanism underneath the observable macroscopic response to mechanical actions. The bonding strength and stiffness 4
existing between atoms entail the response of traditional materials, while the entropic-related effects, typical of elastomers, are mainly responsible for the behaviour of very deformable amorphous materials, since their polymeric chains (connected to each other through covalent bonding in several cross-link junctions, Fig. 1b) can be easily deformed by their geometrical rearrangement. The reduction of the level of high disorder in the entangled polymeric chain network under a mechanical stress increases the material’s internal energy and reduces the corresponding entropic level, so that the two above internal variables (deformation energy and entropy) can be related each other (Fig. 1c).
In summary, the microstructural nature of
polymeric materials entails their physical properties to be mainly governed by the degree of cross-linking existing between the constituting chains.
Please insert here Fig. 1
In the present paper, the problem of defect-induced failure in elastomeric sheets is examined from both experimental and theoretical point-of-view. The influence of the degree of severity of the initial defect and the strain rate effect are both analyzed: in particular, the strain rate influence on the mechanical response is examined to account for the creep phenomenon that has a relevant role in such a class of materials, and can significantly affect their defect-tolerance ability. Experimental tests involving various degrees of severity of the initial defect under different strain rates are interpreted through a theoretical model based on a statistical description of the material at the micromechanical level.
5
2. Some basic concepts of the mechanics of polymers 2.1. Statistics-based description of mechanics of polymers The mechanical response of a polymer network, such as that characterizing an elastomeric material, can be obtained by analyzing the elasticity behaviour of a single polymer chain. A chain is the part of the linear polymeric macromolecule placed between neighboring cross-links (Fig. 1b, c). According to the Kuhn’s theory of rubber elasticity [7-12], typically formulated on the basis of the random walk theory (Fig. 2), the entropy variation S between two states (say reference state 0 and final state 1 , Fig. 1b, c) in a given small volume of the material chain is given by [7, 9]: 1 S k ln( P1 / P0 ) , with Pi (r ) 2 2N
3/ 2
r N r exp 2 N 2
2
(1)
where k 1.38 1023 J/K is the Boltzmann constant, whereas N , r 0, 2 b 2 / 3 are the number of segments in the polymer chain, the average value of the end-to-end distance (with sign) of one chain, and the variance of the chain length (being b the average length of one segment of the chain, Fig. 2(a1)), respectively. The distribution function P i (r ) provides the probability that the end-to-end vector of one single chain is equal to r (Fig. 2b).
Please insert here Fig. 2
6
For a single stretched polymer chain, the variation of the deformation work (F) produced by the stretch acting on the material (quantified by the deformation tensor F ), can be related to its changing of entropy as follows [7]:
(λ ) T S ½ kT 2x 2y 2z 3
(F) ½ kT tr(FT F) 3 ,
(2)
where (λ ) represents the variation of the deformation work (F) in the particular case of a material stretched along only the three coordinate axes by the stretches x , y , z . The energy per single chain given by Eq. (2a) can be used to determine the energy stored per unit volume of material by adding up the contribution of all the single chains contained in such a volume. Once the chain end-to-end distance distribution in the material is known, the overall energy can be evaluated. The chain end-to-end distance distribution in polymers is usually assumed to be described by a Gaussian function, (r, t ) :
3 (r, t ) 2 2Nb
3/ 2
3 r2 c exp c r 2 exp 2 2 Nb 3/ 2
(3)
where the term c represents the cross-links concentration per unit volume. On the other hand, the variation of the deformation energy stored in one single chain when its initial end-to-end distance r0 becomes equal to r Fr0 is given by
U (r )
3kT 2 Nb
2
r2
3kT 2 Nb
2
(Fr ) 0
2
r02
(4)
Finally, by considering Eqs (3) and (4), the variation of the deformation energy stored in the unit volume of polymer can be expressed as follows:
7
V (r, t ) (r, t )U (r) dr
(5)
For a stretched volume of material, whose deformation is quantified by the deformation gradient F , equation (5) can explicitly be written as follows: V (F)
3nkT 2 Nb 2
0 dN (r, N ) (Fr0 )
2
r02 dr
3nkT tr(FT F) 3 (r )dr dr 2 2 Nb Nb 2 / 3
(6)
nkT AkT tr(FT F) 3 tr(FT F) 3 2 2
where it is assumed, as is typical within the rubber elasticity theory, that the deformation process affects only the energy U (r ) per single chain, while it does not affect the distribution function
(r, t ) .
2.2. Interpretation of the effect of deformation on the stored energy The energy variation produced by the deformation occurring in the polymeric network has been written in Eq. (5) by using the energy variation per unit chain given by Eq. (4). On the other hand, the effect of stretching applied to the network’s chains (Fig. 2(a2)) can also be interpreted as a modification of the end-to-end length distribution (r, t ) of the entangled linear polymeric molecules (Fig. 2b). As a matter of fact, due to chains extension, the concentration of links for a given end-to-end distance is not equal to that in the stress-free state [7]. However, the total number of cross-links per unit volume of material does not change because of the stretch (at least if no failure occurs) and, therefore, the new probability distribution function must preserve its area regardless of the amount of the applied stretch (Fig. 2b). The variation of the internal energy according to such an interpretation leads to:
8
V (r, t ) V (r, t ) V (r,0)
t (r, t ) 0 (r)U (r) dr
(7)
In general, the infinitesimal variation of the probability distribution function can be expressed as follows [13]:
dt (r, t ) dt (r, t ) XL dt (r, t ) F t (r, t ) XL t (r, t ) F dt
(8)
where the first term dt (r, t ) XL represents the variation of the distribution function for constant cross-links (i.e. no cross-links are lost or created in the time interval dt ), and t (r, t ) F is the variation of the distribution function due to the variation of the cross-link density but at constant deformation gradient F . In the case of constant cross-links, the variation dt (r, t ) XL can explicitly be expressed by performing a power series expansion of the probability distribution function with respect to the end-to-end distance r , i.e. we have to expand the quantity (r r) . After some calculations, we get [13]:
t (r, t ) XL
t (r) 1 F F r t (r) tr F F 1 r
(9)
On the other hand, if we want to account for the changing of the cross-links in the polymer network, the term t (r, t ) F must be evaluated. The increment of the distribution function
t (r, t ) F for a constant deformation gradient F can also be exploited to simulate the creep phenomenon. It is worth mentioning that the cross-links concentration in the unit volume can be considered
to correspond to the total number of linked chains in the network, i.e. c(t ) (r, t )dr . In the
9
case of no variation of the cross-links numbers per unit volume, such a concentration must be constant during the deformation process, i.e. c(t ) c0 const., t . The above presented formulation for the mechanics of polymeric materials – based on the micro mechanics of the stretched elastomer chains and on the statistical distribution of their endto-end vector – has the advantage of allowing to easily keep into account for large strain features and to consider the possibility for cross-links internal rearrangement (through the term t (r, t ) F in Eq. (8)) that can be exploited to model viscoelastic phenomena; however this last issue will be not addressed in the present study.
2.3. Failure model for polymer chains network Failure of soft elastomers has also been explained through the growth of spherical-like cavities at the mesoscale level, due of the very low elastic modulus compared to the cohesive strength: these mechanical properties are responsible for the (tensile) pressure sensitivity of this class of materials [14-16]. Studies concerning the failure mechanisms in elastomers revealed that microscopic voids typically grow and coalesce into macroscopic ones under hydrostatic tension; sometimes such a process in rubber-like materials is also termed as cavitation [17]. The process of cavity growth and expansion usually exhibits a stiffening behaviour due to unfolding of the long constituents molecules of the polymer [14-17], and other studies have concluded that failure in polymers is driven by fibril creep mechanism [18]. This entails a stable damage phenomenon and justifies, from a qualitative point of view, the defect tolerance in soft polymeric materials. Since the elastomeric materials behave elastically with a good approximation (at least for sufficiently high
10
strain rates), the Griffith’s energy balance for fracture failure referred to a penny shaped circular crack of radius can be written as follows: W G / 0
(10)
where W , , G 4 2 are the potential energy of the external force, the deformation energy stored in the material and the fracture energy, respectively ( being the fracture energy per unit surface). Since the external load is applied statically and the fracture failure is an instantaneous phenomenon, the force potential is zero ( W 0 ) and, therefore, the above energy balance leads to the well-known fracture condition:
/ 4 2
(11)
where the decreasing nature of the internal energy of the material during the fracture process is taken into account. The relationship between the fracture energy and the critical tensile pressure, that produces an unstable expansion of an existing spherical cavity with initial radius in an infinite elastic neo-Hookean medium, can be determined by writing the energy release rate G evaluated between two penny-shaped cracks with radius and + d , respectively [16]: ( ) ( d ) d 0 2d
G lim
(12)
By assuming that the crack can be expanded in a spherical void with the same radius without any further energy, the term ( ) can be evaluated through the solution of the elastic problem related to the case of a spherical cavity in an infinite elastic medium inflated by a pressure p [19]:
11
1 2S 2S1 G 2 E 3
(13)
where the function ( ) has been expressed by making use of the surface stretch S and the Young modulus of the material. The above surface stretch S can be also related to the pressure p acting inside the spherical void through the relation p(S ) E (5 4S1 S4 ) / 6 , with
S S ' / S 1 2d 1 , S , S ' being the external surfaces of the initial and final spherical void, respectively. Equation (13) can be expressed only in terms of the pressure p by substituting the relation p(S ) .
For sufficiently large pressure values, namely p 11E / 18 [16], Eq. (13)
becomes: G
2 E 16 / 25 3 3p 2 3 [1 (6 p / 5E )] 2 E
(14)
By assuming that the energy release rate G is equal to the fracture energy of the material,
G , the critical pressure pc leading to failure can be determined: 2 E 16 / 25 3 3 pc 0, 3 [1 (6 pc / 5E )]2 2 E
with p 5E / 6
(15)
The above pressure corresponds to the hydrostatic stress in the material, i.e. p ii / 3 , and thus the failure criterion for void growth can be stated on the basis of the stress components as follows: 2 E 16 / 25 3 ii 0 3 [1 (2 ii / 5E )]2 2 E
12
(16)
That requires the knowledge of the average radius of the voids which are initially in the material. As soon as the critical condition for void expansion is attained, it can be assumed that the stress state in that point vanishes because of the material failure occurring in the small region around such a point of cavitation.
2.4. Stress state evaluation According to the statistics-based theory of elasticity typically adopted for rubber materials, the knowledge of the chains’ length distribution allows for the evaluation of the internal energy variation with respect to the initial reference state, V (r, t ) . The first Piola stress tensor can be obtained as the gradient of the internal energy variation with respect to the components of the deformation gradient, i.e. P V (r, t ) / F . By using Eqs (7) and (11), the above derivation provides the expression of the first Piola stress tensor and that of the Cauchy stress tensor at a given time t of the deformation process:
P ( F, t )
V (r, t ) F 1r (r, t )F T U (r) dr F r
σ P(F, t ) FT / J
(17a)
(17b)
J det F being the determinant of the deformation gradient. Further, the Green’s strain tensor
can be evaluated through the well-known relationship ε ½(FT F I) [20].
13
3. Numerical implementation of the statistics-based model for polymers failure The above described micromechanics-based model can easily be implemented in a computational tool such as in a model based on the finite element method. The non-linearity of the mechanical response of the material, both in the cases involving no failure – where the material shows a hyperelastic response according to the adopted statistical approach for rubber elasticity – and in the case of chains failure, requires to solve incrementally the problem. Referring to a problem discretized through finite elements, once the internal energy is known for a given applied load or displacement, the first Piola stress tensor P and the Cauchy stress tensor σ can be evaluated at each finite element’s Gauss point through Eqs (17a, b), and used to evaluate the vector Φ e of the unbalanced nodal forces of each finite element :
Φe
B
Ωe
T
σ N b dΩ
N
T
t d Pe 0
e t
(18a)
to be employed in the iterative solution process after assembling the unbalanced nodal force values of each element into the corresponding global vector related to the whole structural component being analyzed. In Eq.(18), e , e t are the current domain and traction boundary configurations of the finite element in turn, B, N, b, t are the compatibility matrix, the matrix of the shape function, the vector of the volume and the vector of the traction forces applied to the finite element’s boundary at the current time instant, respectively. Further, Pe is the nodal vector of the external forces applied to the finite element e . The initial cross-link concentration of the material can be obtained by the well-known relationship c(0)
(r,0)dr E /(3 k T ) that provides a connection between the standard
Young modulus E and the junction density per unit volume of the polymeric network [7, 11]. 14
At every time instant of the loading process, the internal energy V (r, t ) (assessed with respect to the stress-free state through Eq. (7)) is computed after updating the distribution function by means of the expression t (r, t dt ) t (r, t ) dt (r, t ) dt , where the increment dt (r, t ) is evaluated by using Eq. (8). In other words, the Cauchy stress tensor is determined
by the following constitutive relationship (r, t ) σ F 1r (r, t )F T U (r ) dr FT J 1 r
(18b)
that relates σ to the deformation gradient tensor F through some evolving material-related parameters (embedded in the actual distribution function (r, t ) ) and to the energy stored in a single chain given by the standard expression U (r) (see Eq. (4)) arising from the Gaussian statistics of the polymer chains network [7]. The above proposed micromechanics-based formulation of the failure phenomenon does not allow to represent the formation of crack-like ruptures in the material, but it considers the damage as a continuous field variable through the relaxed stress field produced by the chains failure. No discontinuity is allowed to occur and, therefore, the material maintains its continuity. Fully nonlinear geometrical analyses are carried out and, therefore, the integrals (18) are evaluated in the current deformed configuration of the elements in order to account for deformation in the equilibrium states evaluated along the loading path.
4. Experimental tests In order to quantify the mechanical response of flawed elastomeric structural components under tension up to the final failure, different notched thin plates are examined. Pre-notched sheets
15
made of a common silicone polymer obtained from the cross-linking of two vinyl-terminated polydimethylsiloxane matrix and an hydride terminated siloxane curing agent, with a Pdcatalyzed hydrosilylation, also commercially known as Sylgard®, are tested under tension at different strain rates. Then, experimental results are elaborated with Digital Image Correlation (DIC) technique to obtain the deformation field in the samples during the tests. The tested elastomeric sheets have initial elastic modulus equal to about E 0.98MPa and Poisson’s ratio 0.42 , whereas the geometric characteristics of the specimens are shown in Tab.1. In particular, three groups of notched plates have different values of the root radius ( 1.25 mm , 0.312 mm and 0.005 mm ), whereas the last one is a cracked specimen (corresponding to the asymptotic case of a notch with zero curvature radius). In Tab. 1, the geometrical notch parameters a, d , and the corresponding stress concentration factor Kt ( ) for each actual finite plate are reported: such a local stress field parameter is defined as the ratio
Kt max / nom between the maximum to the nominal stresses evaluated at the notch root [21, 22]. In the case of a plate with an elliptical notch with semi-axes a, d (Tab. 1), the minimum curvature at the root of such an elliptical notch profile and the corresponding stress concentration factor in the case of an infinite plate can be expressed as follows:
d2 / a,
K t 1 2a / d 1 2 a /
(19)
In the case of a finite plate with the same notch, the corresponding stress concentration factor can be obtained as Kt f (a / W ) K t , where f (a / W ) is a function of the ratio of notch size to plate size [23]. The use of different values of notch radius enables to understand the effect of the peak stress as well as of the stress gradient on the mechanical response and failure mechanism of highly deformable polymeric materials: as a matter of fact, the local reorganization and the 16
chains re-arrangement in the most stressed region of the body is expected to enable a noticeable defect tolerance capability of the component that should tend to be nearly defect-severity insensitive. The specimens have been tested by imposing an increasing displacement (t ) / 2 , applied with a constant strain rate in the y-direction at each horizontal edge of the plate (see right-hand side of Tab. 1) up to the final failure. Three different strain rates have been applied to the specimens: (i) y1 3.846 103 s 1 , (ii) y 2 9.615 104 s 1 and (iii) y 3 1.603 104 s 1 . These deformation velocities correspond to apply a displacement increment to the silicone sheet equal to = 1mm/2.5s, 1mm/10s, 1mm/60s, respectively. The use of different strain rates enables to examine the effect of relaxation that could eventually occur in such materials: slower strain should allow a better rearrangement of the polymeric network in highly stressed regions, enabling a better defect insensitivity behaviour.
Please insert here Tab. 1
In Figs 3 to 5, the deformed shapes of the specimens for different applied nominal remote strain levels ( y ,0 (t ) (t ) / 2L ) are shown for the three strain rates y1, y 2 , y 3 , applied in the tests. Irrespective of the initial notch severity, the maximum applicable remote strain before failure can be noticed to not depend on the stress concentration factor of the notch. Because of the notch blunting and the local instability occurring close to the notch edges [23], the notch effect results to be similar in all cases, and the initial notch root value does not heavily affect the mechanical behaviour of the sheet. Moreover, note that slower strain rates allow, in general, to get a bigger maximum strain value before failure. 17
Please insert here Fig. 3
Please insert here Fig. 4
Please insert here Fig. 5
Differences in the maximum strain values at failure can be also explained by accounting for the internal micro defects in the polymer. As a matter of fact, the production of the elastomer entails the inclusion of micro air bubbles that have to be removed by an autoclave process. However, such a procedure cannot completely remove such inclusions, and the presence of small bubbles is unavoidable. Note that their detrimental effect is more severe in specimens with blunt notches than in those with sharp notches: in fact, the high stress levels produced by defects with small values of the tip curvature radius can lead to cavitation failure (see Eq. (16)) of spherical inclusions with smaller values of radius . In order to study the effect of the strain rate on the mechanical response of the material, several relaxation tests have been performed. In Fig. 6, the stress-time relaxation tests are displayed when a nominal strain equal to y,0 40% is applied to the material. As can be noted, the strain rate has some slight effect on the stress against time relationships due to the polymer chains internal rearrangement with time. In other words, the chains tend to align more and more along the loading direction while the applied strain is kept on the material. This stress relaxation and chains reordering contribute to reach higher strain values before failure of the material.
18
Please insert here Fig. 6
Failure surprisingly takes place at a level of the remote nominal strain y ,0 that does not reduce with the notch severity, i.e. by increasing the SCF value K t ( ) .
The elevated
deformation of the material allows the stress value in the vicinity of the notch root to remain limited, and the failure condition results to be almost independent of the initial notch radius; further, the complete rearrangement of the notch shape is also responsible for the direction of the crack initiation not to be in pure Mode I (Figs 3 to 5). On the basis of the strain energy at failure (Fig. 7), the fracture energy of the material has been estimated to be equal to about 2.5 104 N / m , whereas the voids radius has been estimated to be approximately equal to 10 4 m . These values are used in the numerical simulations shown in the following section for predicting the cavitation-like failure in the material.
Please insert here Fig. 7
5. Numerical analysis and discussion The theoretical mechanical model for elastomers presented in Sub-Sections 2.2, 2.3 and 2.4 can conveniently be implemented in a FE code by evaluating the stress state through the developed micromechanical approach applied at the Gauss point level. In the present research, the above model has been implemented in a non-linear 2D FE code, and the simulation of the mechanical response of notched elastomeric sheets has been compared with the experimental outcomes. 19
In Fig. 8, the load-displacement curves obtained numerically and from the experimental tests are displayed up to a maximum applied strain equal to about 33%. The stiffness of the elements is correctly predicted by the FE model despite some irregularities in the numerical results (such irregularities are due to the iterative procedure employed).
Please insert here Fig. 8
Figure 9 displays the deformed patterns provided by the FE analyses for plates No.1a and No.4a, for two levels of the remote applied strain y ,0 . The evolution of the deformed shape and the development of the local failure at the notch tip is also provided by the numerical model, where the mechanical response of the polymer has been implemented according to Eq. (17a) and the cavitation-like failure criterion is described by Eq. (16).
Please insert here Fig. 9
Finally, the failure patterns developed during the experimental tests for the specimens No. 1a and 4a are displayed in Fig. 10. As can be observed, the failure modes resemble those predicted by the numerical analyses, that is to say, a nearly dovetail-like failure shape is recognizable.
Please insert here Fig. 10
According to the obtained results, the examined soft polymer shows the ability to withstand macroscopic defects at high applied strain levels, irrespective of the local curvature radius of the considered notch. This unexpected behaviour comes from noticeable local reshape of the defect 20
and from the mechanism of entanglement reduction of the polymeric chains during the stretch process. Moreover, small defects embedded in the polymer, like micro voids, are responsible for the progressive damage in the material which fails due to the expansion and coalescence of the existing micro-bubbles under the tensile hydrostatic stress state. The complex mechanical response of polymers, characterized by anisotropy, time dependent and uncommon damage and fracture behaviour, has been analyzed in Ref. [24] for monotonic and cyclic loading by adopting different material strain-energy density functions (corresponding to a simple hyperelastic model), whereas the effect of strain rate and temperature influence in semicrystalline thermoplastic polymer has been examined in Ref. [25].
It has been
experimentally shown that increasing the temperature or lowering the strain rate entail a softer response of the polymer due to the internal cross-links rearrangement of the material. Such a phenomenon can be quantitatively described by the present model through the term t (r, t ) F in Eq. (8). As far as the damage mechanism is concerned, the above observed behaviour is similar to that of a hierarchical material in which the macroscopic failure proceeds by involving (micro) failures at lower scales as has been shown to occur in several natural materials [26] that can be considered to be optimized with respect to the defect tolerance capability.
6. Conclusions The defect tolerance against failure in highly deformable materials has been examined in the present paper. A theoretical model based on the so-called rubber elasticity has been developed to explain the ability of such a class of material to withstand very high strain values, irrespective of the notch or defect severity.
21
Notched and cracked elastomeric plates made of a silicone polymer have been tested by applying an increasing strain (three different values of strain rate have been analyzed) up to the final failure. The tests have shown that the failure condition is almost independent of the initial stress concentration factor, because the high deformation of the elastomeric sheet around the flawed region smoothes out the peak stress even for an initially sharp straight crack. Then the observed mechanical behaviour is interpreted through a model based on the micromechanics of failure in elastomeric-like materials composed by long entangled crosslinked chains: the proposed approach can macroscopically describe the failure of elastomeric materials on the basis of the micro-damage mechanisms occurring at high strain levels in crosslinked polymeric chains. Strain rate effect has been found to have some influence in the failure behaviour of highly deformable flawed polymer specimens, but it has been neglected in the present theoretical model since we focused mainly on the development of the deformation model and failure for short-time loading for such a class of materials. Further studies and enrichments of the developed theoretical framework are needed. Mechanical failure of soft materials, that are broadly used in many modern applications such as in bioengineering dealing with biological tissues, gels, liquid crystals, etc., can conveniently be examined through the proposed model, which is suitable to be implemented in computational codes.
Acknowledgements The Authors gratefully acknowledge Prof. E. Dalcanale of the Dept. of Chemistry of Parma Univ. for the preparation of the Sylgard® silicone polymer sheet specimens.
22
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[2]
H.M. Westergaard, Bearing pressure and cracks, J. of Appl. Mech. 6 (1939) 49–93.
[3]
M. Creager, P.C. Paris, Elastic field equations for blunt cracks with reference to stress corrosion cracking, Fract. Mech. 3 (1967) 247–252.
[4]
P.G. de Gennes, Reptation of a Polymer Chain in Presence of Fixed Obstacles, Journal of Chemical Physics 55 (1971) 572–579.
[5]
P.G. de Gennes, Soft Matter, Nobel Lecture, Dec. 9, 1991.
[6]
P.J. Flory, Principles of Polymer Chemistry, Cornell University, George Banta Publ. Company, Menasha, Wisconsin, 1953.
[7]
M. Doi, Soft Matter Physics, Oxford Univ. Press, UK, 2013.
[8]
W. Kuhn, Dependence of the average transversal on the longitudinal dimensions of statistical coils formed by chain molecules, J. Polymer Sci. 1 (1946) 380.
[9]
P.J. Flory, Statistical Mechanics of Chain Molecules, Hanser-Gardner, Cincinnati, Ohio, 1989.
[10] L.R.G. Treloar, Physics of Rubber Elasticity, Oxford University Press, 1975. [11] L.R.G. Treloar, The elasticity and related properties of rubbers, Rep. Prog. Phys. 36 (1973) 755–826. [12] E. Guth, H.M. James, Elastic and thermoelastic properties of rubber like materials, Ind. Eng. Chem. 33(5) (1941) 624–629. [13] R. Brighenti, F.J. Vernerey, A simple statistical approach to model the time-dependent response of polymers with reversible cross-links, Composite Part B: Eng. (2016), in press (DOI: 10.1016/j.compositesb.2016.09.090). [14] Y.Y. Lin, C.Y. Hui, Cavity growth from crack-like defects in soft materials, J. Fract. 126(3) (2004) 205–221. [15] C. Fond, Cavitation criterion for rubber materials: a review of void-growth models, J Pol. Sci.: Part B: Polymer Physics 39 (2001) 2081–2096.
23
[16] Y.Y. Lin, C.Y. Hui, Cavity growth from crack-like defects in soft materials, J. Fract. 126 (2004) 205–221. [17] Y. Lev, K.Y. Volokh, On cavitation in rubberlike materials, J. App. Mech. 83(4) (2016) 0044501-1–0044501-4. [18] M. Jie, C.Y. Tang, Y.P. Li, C.C. Li, Damage evolution and energy dissipation of polymers with crazes, Theor. Appl. Fract. Mech. 28(3) (1998) 165–174. [19] A.N. Gent, C. Wang, Fracture mechanics and cavitation in rubber-like solids, J. Mat. Sci. 26(12) (1991) 3392–3395. [20] G.A. Holzapfel, Nonlinear solid mechanics: a continuum approach for engineering, John Wiley & Sons, West Sussex, England, 2000. [21] W. C. Young, R. G. Budynas, Roark’s formulas for stress and strain, McGraw-Hill, 7th edition, 2002. [22] W. D. Pilkey, D. F. Pilkey, Peterson's Stress Concentration Factors, 3rd Edition, Wiley, 2008. [23] R. Brighenti, A. Spagnoli, A. Carpinteri, F. Artoni, Notch effect in highly deformable material sheets, Thin-Walled Structures 105 (2016) 90–100. [24] D.A. Serban, L. Marsavina, V. Silberschmidt, Behaviour of semi-crystalline thermoplastic polymers: Experimental studies and simulations, Comput. Mat. Sci. 52 (2012) 139–146. [25] D.A. Serban, G. Weber, L. Marsavina, V.V. Silberschmidt, W. Hufenbach, Tensile properties of semi-crystalline thermoplastic polymers: Effects of temperature and strain rates, Polymer Testing 32 (2013) 413–425. [26] R. Mirzaeifar, L.S. Dimas, Z. Qin, M.J. Buehler, Defect-tolerant bioinspired hierarchical composites: simulation and experiment, ACS Biomater. Sci. Eng. 1 (2015) 295−304.
24
Figures and captions
(a)
0
1
(b)
(c)
Fig. 1. (a) Scheme of a 3D polymer network. Simple sketch of : (b) unstretched configuration 0 and (c) stretched configuration 1 of the cross-linked polymeric chains.
25
Chains microscopic end to end distance density probability
b
z
b r0
(a1)
y
O
x
F
z
r
(a2)
F=
[
(b)
(0)
(
y
O
]
0(
0
stretch ( )
Chains end-to-end distance, r =
x
Fig. 2. (a1) Scheme of an undeformed polymer chain and definition of its end-to-end vector; (a2) corresponding deformed chain; (b) probability distribution functions of the chains lengths before and after deformation.
Spec. No.
1a
W
2a
(mm) (mm) 114 40
2d
t
(mm) (mm) 10 2.10
2a/W
a/d
(---) (---) 0.351 4
Kt ( )
(mm) (---) 1.250 12.43
y (s-1)
y1
1b
114
40
10
1.50
0.351
4
1.250 12.43
y 2
1c
114
40
10
1.95
0.351
4
1.250 12.43
y 3
2a
114
40
5
1.50
0.351
8
0.312 24.67
y1
2b
114
40
5
2.25
0.351
8
0.312 24.67
y 2
2c
114
40
5
2.25
0.351
8
0.312 24.67
y 3
3a
114
40
2
1.60
0.351
20
0.005 61.73
y1
3b
114
40
2
2.00
0.351
20
0.005 61.73
y 2
3c
114
40
2
1.80
0.351
20
0.005 61.73
y 3
4a
114
40
<0.05
2.15
0.351 >400
~0 26
---
y1
4b
114
40
<0.05
1.80
0.351 >400
~0
---
y 2
4c
114
40
<0.05
2.00
0.351 >400
~0
---
y 3
Tab. 1. Geometric characteristics of the tested specimens and related stress concentration factors K t ( ) . Specimens labeled as na, nb and nc (with n=1, 2, 3, 4) correspond to the strain rates y1, y 2 , y 3 , respectively.
27
Fig. 3. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y1 3.846 103 s 1 strain rate.
28
Fig. 4. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y 2 9.615 104 s 1 strain rate.
29
Fig. 5. Images of the specimens in the initial ( y,0 0.0 , a, d, g, j), intermediate ( y,0 33% , b, e, h, k) and final configuration at incipient failure (various y ,0 values, c, f, i, l). The first, second, third and fourth row correspond to the specimen type 1, 2, 3, 4, respectively, tested at
y3 1.603 104 s 1 strain rate.
30
3.6E+005
2 nd cycle
y,0= 0.4
0
(b)
3 rd cycle
2.8E+005 100
3rd cycle
1 st cycle 3.0E+005
1st cycle
3.2E+005
2nd cycle
1
factor, f (t)
Stress, x (Pa)
(a) 3.4E+005
200
300
400
500
0
600
600
1200
1800
2400
3000
Time, t (s)
Time, t (s)
Fig. 6. (a) Stress relaxation test under an applied nominal strain equal to y,0 40% . (b) Three cycles are examined, namely y (t ) f (t ) y,0 .
(a1)
(a2) Force, F (N)
failure
spec. 1a spec. 2a
200 150 100 50
(b) 0 0
10
20
30
40
Displacement, (mm)
50
60
Fig. 7. Crack paths at failure in the specimen 1a (a1) and 2a (a2); (b) load-displacement curve for the same specimens up to the final fracture collapse.
31
Force, F (N)
Applied force, F (N)
(a)
40
spec. No. 1a 30 20 exp. results FE results
10
(b)
40
spec. No. 4a 30 20 exp. results FE results
10 0
0 0
5
10
15
Applied displacement, (mm)
20
0
5
10
15
Applied displacement, (mm)
20
Fig. 8. Experimental and numerical force F against applied displacement (see Tab. 1) for: (a) the notched specimen 1a ( 2b 10mm ), and (b) the cracked specimen.
(a1)
(b1)
(a2)
(b2)
Fig. 9. Deformed patterns and Green’s strain Eyy determined by the model of failure for elastomers implemented in a FE code. Specimens No. 1a (a1, a2) and No. 4a (b1, b2), for a remote nominal applied strain y,0 18% (a1, b1) and y,0 33% (a2, b2). Details of the local failure patterns are also reported in the two magnifications.
32
(a)
(b)
Fig. 10. Detail of the failure pattern at the notch tip for specimens 1a (a) and 4a (b), developed during the experimental tests.
33
Highlights
- Failure defect sensitivity in polymers is examined - A theoretical mechanical model is formulated based on entropy concept - A theoretical failure mechanism of polymers is taken into account - Failure experimental results of notched sheets are presented - Experimental results are compared with numerical outcomes
34