journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
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Influence of initial flaws on the mechanical properties of nacre S. Anup Indian Institute of Space science and Technology (IIST), Valiamala, Thiruvananthapuram 695547, India
ar t ic l e in f o
abs tra ct
Article history:
Nacre is a bio-composite made up of hard mineral and soft protein, and has excellent
Received 20 January 2015
mechanical properties. This paper examines the effect of naturally occurring defects
Accepted 23 February 2015
(initial flaws) in nacre on its mechanical properties such as toughness and strength. A
Available online 5 March 2015
random fuse model is developed incorporating initial flaws. Numerical simulations show
Keywords:
that initial flaws affect different mechanical properties at different rates. The variation in
Random fuse model
the experimentally obtained mechanical properties of nacre reported in the literature is
Defect
shown to be due to initial flaws. The stress in the mineral and protein increases due to
Nacre
initial flaws, but by different amounts. The results obtained in this study are useful for
Biocomposite
gaining insight into the failure of nacre and development of nacre-inspired composites.
Mechanical properties
1.
Introduction
Biological materials such as bone and nacre are composites of soft organic protein and hard, brittle mineral. They possess interesting features and properties. The toughness of these bio-composites is excellent, especially when compared to their very weak constituents. Consider the case of nacre. This is the inner layer of the mollusk shell (Shao et al., 2012). At the micro-scale, nacre consists of aragonite (calcium carbonate) platelets arranged in a staggered fashion. An organic matrix material fills the space between the platelets. This arrangement is referred to as the “Brick and Mortar structure” (Gao et al., 2003; Katti and Katti, 2006). The organic matrix consists of domains, which unfold one after the other without causing the molecular backbone to break (Smith et al., 1999). This leads to a saw tooth type force deflection curve for the organic matrix with a very large deformation. The fracture toughness of nacre is about 3–7 MPa m 1=2 , though the constituent, mineral has a fracture toughness much less than 1 MPa m1=2 (Ji and Gao, 2004). Many studies were conducted to examine the reasons for the excellent toughness of nacre (Okumura and de E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.jmbbm.2015.02.026 1751-6161/& 2015 Elsevier Ltd. All rights reserved.
& 2015 Elsevier Ltd. All rights reserved.
Gennes, 2001; Jackson et al., 1988). Barthelat and Rabiei (2011) conducted experiments and found out that there is a large fracture process zone around the crack tip, enabling high energy dissipation and thereby providing high resistance to fracture. Nukala and Simunovic (2005) found out that domain unfolding plays an important role in the toughness of nacre. A number of defects are present in the microstructure of nacre (Barthelat and Espinosa, 2007). In this paper, we refer to the defects as initial flaws. The defects that could occur include platelet breakage and matrix flaws. However, nacre is supposed to be defect-tolerant (Huang and Li, 2013). Many researchers have investigated the reasons for the flaw tolerance of nacre (Barthelat et al., 2007; Wang et al., 2001; Huang and Li, 2013). Though this is the case, studies about the influence of existing defects on the mechanical properties are not available in open literature, especially how the various types of initial flaws affect the strength and toughness. There is a need to systematically study the effect of initial flaws in the matrix and platelet on the mechanical properties such as strength and toughness of nacre. The understanding of how existing defects affect failure, would be useful not only for
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
getting insight into the failure of nacre, but also for developing nacre-inspired artificial materials. Discrete lattice models have been proposed for analysis of deformation and fracture of materials. In a discrete lattice model, the continuum is discretised into one-dimensional elements. The random fuse model is a special case of a discrete lattice model where these one dimensional elements are modelled as resistors (de Arcangelis et al., 1985; Kahng et al., 1988). In a random fuse model, displacement and force are replaced by their electrical analogues; voltage and current. In short, a resistor network replaces the continuum. The random fuse model (RFM) is a scalar analogue of an elastic discrete lattice model. The solution of an electrical network (Zienkiewicz, 1971) is very similar to that of an elastic network. The random fuse model provides a simpler way of solving the discretised continuum. This is because of the scalar nature of the equations used in RFM. Moreover, RFM has only one degree of freedom per node per element. However, the corresponding spring and beam elements have higher degrees of freedom (Skjetne et al., 2001). The random fuse model can incorporate disorder much more easily than finite element models (Alava et al., 2006). RFM has been extensively used to simulate the mechanical behaviour of materials, especially when randomness in the properties and structure is to be introduced in the model (Hansen, 2005; Sahimi, 2003; Herrmann and Roux, 1990). In order to account for evolving failure of materials, a continuous damage random fuse model (CDRFM) has also been proposed (Zapperi et al., 1997). CDRFM has been employed to model domain unfolding in failure simulation of biological composites (Nukala and Šmunović, 2005; Anup et al., 2008). However, in these models, the effect of initial flaws has not been considered. In this paper, we develop a 2D CDRFM model taking into account initial flaws. Initial flaws of the platelet and the matrix are introduced separately into the model. Numerical solution of the model gives the stress–strain response of the composite. Mechanical properties are derived from this response and the effect of initial flaws on these mechanical properties are examined.
Lm
Gap
169
Wp
Mineral Platelet
Organic Matrix
Fig. 1 – Schematic diagram showing the structure of nacre. Note the staggered arrangement of the mineral platelets in the organic matrix.
Shear Matrix Elements
Platelet Elements
Tensile Matrix Elements
Fig. 2 – A part of the CDRFM model showing the various types of elements. Tension elements are used to represent mineral platelets. Shear elements represent matrix connecting the vertical faces of the platelet; tensile matrix elements connect horizontal faces of the platelets.
edges of platelets in the gap region. However, stresses carried by the matrix connecting these horizontal edges of platelets are negligible (Ji and Gao, 2004; Jäger and Fratzl, 2000; Nukala and Šmunović, 2005). Therefore, these tensile matrix ele-
2.
Formulation of the problem
In order to develop a model for failure of nacre, CDRFM is employed. Fig. 1 shows a schematic structure of nacre with the mineral platelets and organic matrix. The model used for simulation is based on this structure and is similar to that employed by other researchers (Nukala and Šmunović, 2005; Anup et al., 2008). A square lattice network of L L size is used. A part of the network used is shown in Fig. 2. In CDRFM, a current–voltage analogue of the elastic model is employed (Zapperi et al., 1997; Nukala and Simunovic, 2005). Ji and Gao (2004) have developed a tension-shear chain model to explain the stress transfer in nacre-like composites. In this model, the mineral platelets are assumed to carry only tensile forces. The matrix is assumed to transfer shear forces between platelets. Based on this model, we assume that platelet elements carry tensile stresses and matrix elements carry shear stresses. These elements are shown in Fig. 2. Tensile matrix elements could be used to connect the horizontal
ments are given zero stiffness values. In this model, all dimensions are normalised with respect to the width of the platelet, Wp (see Fig. 1). A unit width is assumed in the out of plane direction. Therefore, the area of the platelet, Ap is equal to unity. The platelet is divided into a number of elements equal to the normalised length of the platelet, so that the normalised length of each platelet element is unity. A matrix element has a length equal to Lm as shown in Fig. 1. The width of the matrix element is equal to the length of the matrix element. Therefore, the crosssectional area of a matrix element is also equal to one. Stiffness (conductance) of each element is given by Ce ¼ Ae Ee =Le , where Ee is Young's Modulus, Ae is the area, and Le is the length of the element. The stiffness of a platelet element is equal to Young's modulus, Ep since both area and length are unity. The stiffness of a matrix element is found out to be Cm ¼ Gm =Lm , where Lm is the normalised length of a matrix element and G is the shear modulus of the matrix (Nukala and Šmunović, 2005).
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
The platelet elements are assumed to be linear elastic. The simulation results of failure of nacre (Nukala and Simunovic, 2005) indicate that allowing platelets to fail, results in a very
We assume a random distribution of initial flaws. It is also assumed that the size of each individual initial flaw is that of a single element of platelet or matrix.
small toughness compared to that of the case without platelet breakage. Such a small toughness is contrary to experimental evidence. Therefore, similar to these studies
3.
Numerical simulations
(Nukala and Simunovic, 2005), platelets are not allowed to fail in this study. The protein, being a bio-polymer, has a sawtooth stress–strain response (Smith et al., 1999). This beha-
A system size of 256 256 is employed for the simulation. The mechanical properties of the platelet and matrix given as an input
viour is due to domain unfolding and is incorporated in the 200
the matrix is assumed to evolve with failure. Therefore, this
180
response is represented by shear fuses with conductance
160
decreasing in discrete steps. When the current exceeds the
140
threshold value (strength) of the matrix elements, matrix element is assumed to fail (domain unfolding), and is termed matrix breakage. In order to model this domain unfolding, the stiffness of the matrix element is reduced by multiplying
Stress (MPa)
model using CDRFM (Nukala and Šmunović, 2005). In CDRFM,
120 100
by a factor of a, where 0oao1. Thus, the conductance of the
60
i
matrix element can be calculated to be Cm a , where i is the
40
number of times it has failed. However, when the maximum
20
shear strain of the matrix is reached, the failure of the
0
molecular backbone (matrix cracking) occurs and the con-
0
0.5
1
1.5
2
2.5
3
3.5 −3
x 10
Strain 200 180 160 140 Stress (MPa)
ductance of the element is made zero. Along the left and right edges of the model, periodic boundary conditions are imposed. A constant voltage difference is assumed between the top and bottom of the model. Matlab package (MathWorks, 2013) is used to solve Kirchoff's equations. Numerically, a unit voltage difference is applied between the top and bottom of the model. The ratio (r) of current in each resistor to the threshold value of the resistor is found out. The element having a maximum value of this ratio fails, and consequently, the conductance matrix is modified. The equation being linear, the external current and voltage are scaled by 1=r, so that the current in the element having maximum r reaches the threshold value. In effect, the applied voltage is increased or decreased so that one fuse alone is broken. The modified conductance matrix is used in the next step for the solution of the modified system. This process of breaking of bonds continues until the lattice separates in two parts. The system current ðIc Þ and the system voltage ðVc Þ of the whole composite are found out each time the equations are solved. The ratio of Ic =L to Vc =L gives the stress–strain behaviour of the composite (Nukala and Simunovic, 2005). The maximum value of stress is considered to be the strength of the composite. The toughness is calculated as the area under the stress–strain curve.
80
120 100 80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
3.5 −3
x 10
Strain
Fig. 3 – Comparison of stress–strain responses for varying values of initial (a) platelet flaws and (b) matrix flaws. The curves shown are for initial flaws of 0% (top-blue), 1%, 2%, 5%, 10%, 15%, and 20% (bottom-dark brown). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
Table 1 – Dimensions of nacre at micro-scale: lengths in nm, area in nm2 (Nukala and Simunovic, 2005; Song et al., 2003). Dimension
Value
Platelet width ðW p Þ
500
Gap Platelet length ðLm Þ Matrix area Matrix length
500 3500 500 25
Normalised value
Number of elements
1
1
1 7 1 1/20
1 7 1 1
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
for simulation are similar to that of the constituents of nacre. Thus, the platelet Young's modulus is assigned a value of 100 GPa (Jackson et al., 1990) and shear modulus of matrix is given a value similar to that of the organic matrix, 4.6 GPa (Jackson et al., 1990, 1988). A typical value of 46 MPa (Jackson et al., 1988; Nukala and Simunovic, 2005) is given for the shear strength of the organic matrix, τs. The maximum shear strain that the protein in nacre can undergo is 200% (Wang et al., 2001). In the present work, a value of 200% is also taken to be the maximum shear strain, ϵmax that the matrix is able to undergo. Each time a matrix element breaks, its stiffness values are decreased to 90% of the original value (a ¼ 0:9Þ in order to simulate the breakage of bonds. The maximum number of breakages for each matrix element, nmax is calculated based on the maximum strain and decrease in stiffness (Nukala and Simunovic, 2005), and is given by nmax ¼ ðlogðϵ0 Þ= ðϵmax ÞÞ=logðaÞ 50, where ϵ0 is found from the relation, τs ¼ ϵ0 Gm . Table 1 gives the dimensions of the model used in the present study. These dimensions are similar to that of the structure of nacre (Nukala and Simunovic, 2005). Table 1 also shows the number of elements used for a platelet and matrix. The effect of initial flaws in the platelet and matrix separately on the stress–strain response is investigated in the present study. The sites of the initial flaws is taken to be at random among all the respective elements. The elements existing at these sites are removed before the start of the simulation. We define, percentage initial platelet flaws, DiP ¼ RP =TP 100, where RP is the number of initially removed platelet elements and TP is the total number of platelet elements in the model. The percentage of initial matrix flaws is also defined in a similar way. A parametric study is conducted by varying the percentage of initial flaws from 0 to 20%. Fig. 3(a) shows the stress–strain response for the case of initial platelet flaws; (b) shows the response in the case of initial matrix flaws.
4.
Discussions
There is a change in the stress–strain response when the percentage of initial flaws is varied as shown in Fig. 3(a) and (b). The stiffness, strength, and toughness of the composite are extracted from the stress–strain response. The overall mechanical properties obtained should not depend on system size, and the system size should be large to avoid any effects due to random variations in locations of flaws. Here, we use a large system size of 256 256. Also, in order to obtain statistically correct values, a number of five runs are made for each value of initial flaws. The results thus obtained for the case without flaws are compared between system sizes of 128 128 and 256 256. The stiffness values of the composite obtained for the two system sizes are 73.52 and 73.48 MPa, and the value of strength for the two system sizes are 182.9 and 182.9 MPa. Thus, the difference in these mechanical properties is very small when the system size is doubled from 128 to 256. Therefore, a system size of 128 128 could even be used. However, we report values for a system size of 256 256, for better accuracy. The tensile modulus and strength of nacre for the case without flaws is compared with the works of Kotha et al. (2001) and Barthelat and Rabiei (2011). The analytical expressions for stiffness and strength of the composite from those
research works are given in the following equations: ! 1 1 1 2 ð1 þ coshðγρÞÞ ¼ 1þ þ Ecomp h Ep ρð1=hÞ Gγm sinhðγρÞ σ comp ¼ ρτs =2
171
ð1Þ ð2Þ
Here, Ecomp and σ comp are the stiffness and strength of the composite, ρ is the aspect ratio of the platelet and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ ½Gm =Ep Þð1=hÞ. Taking ρ ¼ 7, Ep ¼100 MPa and mineral volume fraction ϕ ¼ 20=21, gives the composite strength as 161 MPa and stiffness, Ecomp ¼82.1 GPa. In comparison, the results of the simulation give values of 182.9 MPa and 73.48 GPa. The values are comparable, but the difference could be due to the fact that in the derivation of the analytical expressions, matrix shear stress is assumed to be a constant. In contrast, these stresses are allowed to vary in RFM. Note that for the case without flaws, the results obtained are same as that of Nukala and Simunovic (2005). The stiffness (Young's modulus) and strength obtained agree with the experimental results as well (Jackson et al., 1988). The variation of the mechanical properties with initial flaws is shown in Fig. 4(a)–(c). The mechanical properties of the composite are normalised with respect to the case without any flaws. All mechanical properties decrease with an increase in both types of initial flaws, though at varying rates. The stiffness (defined as the initial linear part of the stress– strain response) decreases at about a constant rate. The strength and toughness decreases rapidly initially, but this rate of decrease becomes lesser with an increase in the percentage of initial flaws. For all mechanical properties, it is seen that platelet flaws are the most detrimental. Matrix flaws have a lesser effect on mechanical properties. Fig. 5(a) and (b) compares how different mechanical properties change as a function of initial platelet flaws and initial matrix flaws. The graphs of mechanical properties shown here are the same as in Fig. 4(c), but are grouped in terms of the type of flaw. It is seen from Fig. 5(a) and (b) that for all types of initial flaws, stiffness has the least change, followed by strength. Toughness shows the highest variation with an increase in initial flaws. Fig. 5(a) and (b) also shows the number of domains unfolded (matrix breakage) for different percentages of initial flaws. These figures also show that when the number of broken matrix elements decreases, the strength and toughness also decrease. The number of broken matrix elements decreases with both types of initial flaws. For the same percentage of initial flaws, initial matrix flaws produce a larger number of broken elements than initial platelet flaws. The trends of decrease in toughness and the number of broken matrix elements are similar. The number of breakages decreases as flaws increase, thereby reducing mechanical properties such as toughness. In the simulation, the failure path consists of fully failed (cracked) matrix elements. Cracked matrix elements denote those elements that have reached their maximum strain and have their conductance set equal to zero. We examine how the failure paths vary with respect to the type and number of flaws. The tortuousness of the failure path is defined by the number of failed elements. Let Ni denote the number of failed (cracked) matrix elements. Fig. 6(a) shows a part of the failure path when no initial flaws are present. The failure path is similar to that
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
1 1
Initial Platelet Flaws Initial Matrix Flaws
0.9 0.8
0.9
Normalised Strength
Normalised Stiffness
0.95
0.85 0.8 0.75 Initial Matrix Flaws Initial Platelet Flaws
0.7
0.7 0.6 0.5 0.4 0.3 0.2
0.65
0.1 0
5
10
15
0
20
0
5
10
15
20
Initial Flaws (%)
Initial Flaws (%)
1 0.9
Initial Matrix Flaws Initial Platelet Flaws
Normalised Toughness
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
Initial Flaws (%)
Fig. 4 – Comparison of (a) stiffness (b) strength and (c) toughness for different percentages of flaws. These mechanical properties decrease with an increase in the percentage of initial platelet and matrix flaws.
obtained by Nukala and Simunovic (2005). Fig. 6(b) shows the number of failed elements for both types of initial flaw: platelet and matrix. The failure path is tortuous the most when no initial flaws are present (Ni ¼ 1048). The path tortuousness in descending order is for the cases of initial matrix and initial platelet flaws. Path tortuousness decreases with an increase in percentage of the initial flaws. Therefore, the decrease in toughness for both types of initial flaws (platelet and matrix) could be associated with the lessening of path tortuousness. Several experiments were conducted by different research groups to find out the mechanical properties of nacre (Jackson et al., 1988; Bekah et al., 2012). However, the values reported show a wide variation. The largest variation is for toughness, followed by that of tensile strength and the least for stiffness. For example Jackson et al. (1988) gives values of 350–1240 J/m2 for toughness, 70–140 MPa for tensile strength and 60–70 GPa for stiffness. Another research group reported values of 100–1500 J/m2 for toughness, 70–100 MPa for tensile strength and 70–80 GPa for stiffness (Bekah et al., 2012). The results of the current study show a similar trend in the variation of mechanical properties; toughness the largest variation, stiffness the least and tensile strength in between, for both types of initial flaws (for example, see Fig. 5(a)).
Therefore, this variation in mechanical properties reported in literature could be due to naturally occurring defects in nacre. In the simulations reported in Section 3, the platelets were assumed not to fail. Here, this assumption is critically examined. The maximum stress in the platelets, over the whole simulation, σs for case without flaws is found to be 366 MPa. Since, the platelets in nacre have a strength E400 MPa, the assumption seems to be correct. Researchers have used analytical models to predict the relationship between the tensile stress in platelets and shear stress in matrix, σ ¼ ρ τ (Ji and Gao, 2004), where ρ is the aspect ratio of the platelet. This gives a value of σ s ¼ 7 46 ¼ 322 MPa, which is comparable to that of the present study. The difference in the values could be due to the assumption of constant shear stress made in the analytical model. In contrast, the RFM distributes stress in the matrix to attain equilibrium, and in the process permits variation in shear stress to be developed. Further, we examine the effect of initial flaws on the maximum tensile stress in the platelet over the whole simulation. Fig. 7 shows the variation of this stress with the percentage of flaws. The value of maximum tensile stress decreases with an increase in initial platelet and matrix flaws. This means that if platelet breakage does not happen for the case without flaws, this will not occur for the case
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
Normalised mechanical properties/ Normalised no. of broken elements
1.2
Normalised Stiffness Normalised Strength Normalised Toughness Normalised No. of Broken Elements
1
0.8
0.6
0.4
0.2
0
0
5
10 Initial Platelet Flaws (%)
15
20
1
Normalised mechanical properties/ Normalised no. of broken elements
0.9 0.8 0.7 0.6 0.5 0.4
Normalised Stiffness Normalised Strength Normalised Toughness Normalised No. of Broken Elements
0.3 0.2 0.1 0
0
5
10
15
20
Initial Matrix Flaws (%)
Fig. 5 – Comparison of normalised mechanical properties and normalised number of broken elements for various percentages of initial (a) platelet flaws and (b) matrix flaws. The curves shown are the same as in Fig. 4(a)–(c), but are grouped in terms of the type of flaws.
with initial platelet or matrix flaws. This fact could be made use of in the design of bio-inspired composites. However, note that even with fully matrix failure, these composites with initial flaws has lesser toughness and strength than that of the case where no flaws are present. The above discussion shows that flaws influence the stresses in the elements, and thereby changes the way the elements break. In order to understand how initial flaws influence the stress distribution in the model, we compare the stresses developed in all the elements in the entire model, before failure of any of the elements occurs. Fig. 8(a) shows the stress distribution for the case without any flaws, and (b) and (c) show the stresses when matrix and platelet flaws of 20% are present. Here, the subscripts 0, 1 and 2 denote the following cases: no initial flaws, initial matrix, and initial platelet flaws respectively. All models (without flaws, and with initial matrix and platelet flaws) are subjected to the same displacement, and stresses developed in the elements are found out using RFM. The stresses are normalised with respect to the maximum shear stress in the model for the case without flaws. From 8(a) it can be seen that the stresses in the model are in some specific values when there are no flaws. In contrast, stresses are distributed over a range of values when flaws are present (see Fig. 8(b) and (c)). For the case without flaws, ratio of the maximum tensile stresses in the platelet to that of the maximum shear stress in the matrix, R 5. However, when there is initial flaws (20%) in the matrix, the matrix shear stresses increase by a larger amount than that of the platelet tensile stresses, making this ratio R 2:2, as seen in Fig. 8(b). A similar pattern of increase in the stresses in the platelet and matrix is observed for the case of initial flaws (20%) in the platelet, and R 1:7 (refer Fig. 8(c)) in this case. Therefore, due to initial flaws, though there is an increase of the tensile stress in the platelets, there is a higher increase of shear stress in the matrix. Considering the whole simulation, since the matrix failure occurs at the shear strength of the matrix, this stress distribu-
1100
No. of Cracked Elements
1000
Initial Matrix Flaws Initial Platelet Flaws
900 800 700 600 500 400 300 200
0
5
10
15
20
Initial Flaws (%)
Fig. 6 – (a) Model showing a part of the failure path (cracked matrix elements) in the case of no initial flaws and (b) variation of number of cracked matrix elements with the percentage of initial flaws. The number of cracked matrix elements decreases with an increase in initial flaws.
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
tion means that matrix failure becomes increasingly likely for the case with flaws, when compared to that without. Thus, this matrix failure prevents higher stresses being developed in the
platelet. Consequently, maximum stress in the platelets over the whole simulation is not increased due to flaws. However, note that the stresses shown in Fig. 8(a)–(c) are before breakage of any element. As the simulation proceeds, matrix elements break, changing the stress in all the elements after each break. In order to develop bio-inspired materials, following could be the suggestions based on the results of this work
1 0.98
Normalised Stress
0.96 0.94 0.92
1. Bio-inspired composites could obtain better mechanical properties than that of nacre by eliminating or reducing initial flaws. 2. Priority should be given to reduce initial platelet flaws than matrix flaws to get highest improvement in mechanical properties.
Initial Platelet Flaws Initial Matrix Flaws
0.9 0.88 0.86 0.84 0.82 0
5
10
15
In the present study, we assume that matrix at the end of the platelets could not transfer any load. However, some studies indicate that the matrix connecting these short platelet edges could influence the stress distribution in platelets (Bekah et al., 2012). The influence of such tensile elements will be incorporated in future studies. We used the random fuse model in this work to find the effect of flaws on the overall mechanical
20
Initial Flaws (%)
Fig. 7 – Variation of maximum tensile stress in the platelet, for the case with initial flaws. Stress is normalised with respect to that of the case without flaws. Maximum tensile stress decreases with an increase in the number of flaws.
7
5
6
4
τ0 / τ0max
3.5
σ 0 / τ0max
τ1 / τ0max σ 1 / τ0max
5 Normalised Stress
Normalised Stress
4.5
3 2.5 2 1.5
4 3 2
1
1 0.5 0
0
2
4
6
8
10
12
0
14
0
2
4
6
8
4
10
12
14 4
x 10
Element Number
Element Number
x 10
10 9
τ /τ 2
8
2
Normalised Stress
0max
σ /τ
0max
7 6 5 4 3 2 1 0
0
2
4
6
8
10
12
14 4
Element Number
x 10
Fig. 8 – Comparison of the stresses in the model before any breakage of elements, when subjected to the same displacement at the boundary: (a) no initial flaws; (b) initial matrix flaws (20%); (c) initial platelet flaws (20%).
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
properties. However, other tools such as finite element methods (FEM) do have certain advantages over the RFM. In FEM, an extensive analysis of the stress distribution in the platelet and matrix near the sites of flaws could be undertaken. These analyses would form part of the future study.
5.
Conclusions
A continuous damage random fuse model for nacre was developed which takes into account the effect of initial flaws in the platelet and the matrix on the composite mechanical properties. The following are the major conclusions.
All mechanical properties show a decrease with an
increase in initial flaws, though at varying rates. Stiffness decreases at about a constant rate. All other mechanical properties show a rapid decrease for lower values of initial flaws, but with further increase in initial flaws, rate of decrease reduces. The mechanical properties most influenced by initial flaws in descending order are toughness, strength, and stiffness. Decrease in toughness due to the presence of initial damage could be attributed to the decrease in the number of unfolded domains and reduction in tortuousness of the failure path. Variation in the reported experimental results of mechanical properties could be due to initial flaws present in nacre. Initial flaws cause higher increase in shear stress in the matrix than in the platelet.
These results could be important in understanding how nacre achieves its mechanical properties, especially toughness. The information obtained would also be useful in the design of bio-inspired composites.
Acknowledgement The author would like to thank Dr. Arun C.O. and Dr. Praveen Krishna I.R. of IIST for useful discussions and suggestions.
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Influence of initial flaws on the mechanical properties of nacre S. Anup Indian Institute of Space science and Technology (IIST), Valiamala, Thiruvananthapuram 695547, India
ar t ic l e in f o
abs tra ct
Article history:
Nacre is a bio-composite made up of hard mineral and soft protein, and has excellent
Received 20 January 2015
mechanical properties. This paper examines the effect of naturally occurring defects
Accepted 23 February 2015
(initial flaws) in nacre on its mechanical properties such as toughness and strength. A
Available online 5 March 2015
random fuse model is developed incorporating initial flaws. Numerical simulations show
Keywords:
that initial flaws affect different mechanical properties at different rates. The variation in
Random fuse model
the experimentally obtained mechanical properties of nacre reported in the literature is
Defect
shown to be due to initial flaws. The stress in the mineral and protein increases due to
Nacre
initial flaws, but by different amounts. The results obtained in this study are useful for
Biocomposite
gaining insight into the failure of nacre and development of nacre-inspired composites.
Mechanical properties
1.
Introduction
Biological materials such as bone and nacre are composites of soft organic protein and hard, brittle mineral. They possess interesting features and properties. The toughness of these bio-composites is excellent, especially when compared to their very weak constituents. Consider the case of nacre. This is the inner layer of the mollusk shell (Shao et al., 2012). At the micro-scale, nacre consists of aragonite (calcium carbonate) platelets arranged in a staggered fashion. An organic matrix material fills the space between the platelets. This arrangement is referred to as the “Brick and Mortar structure” (Gao et al., 2003; Katti and Katti, 2006). The organic matrix consists of domains, which unfold one after the other without causing the molecular backbone to break (Smith et al., 1999). This leads to a saw tooth type force deflection curve for the organic matrix with a very large deformation. The fracture toughness of nacre is about 3–7 MPa m 1=2 , though the constituent, mineral has a fracture toughness much less than 1 MPa m1=2 (Ji and Gao, 2004). Many studies were conducted to examine the reasons for the excellent toughness of nacre (Okumura and de E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.jmbbm.2015.02.026 1751-6161/& 2015 Elsevier Ltd. All rights reserved.
& 2015 Elsevier Ltd. All rights reserved.
Gennes, 2001; Jackson et al., 1988). Barthelat and Rabiei (2011) conducted experiments and found out that there is a large fracture process zone around the crack tip, enabling high energy dissipation and thereby providing high resistance to fracture. Nukala and Simunovic (2005) found out that domain unfolding plays an important role in the toughness of nacre. A number of defects are present in the microstructure of nacre (Barthelat and Espinosa, 2007). In this paper, we refer to the defects as initial flaws. The defects that could occur include platelet breakage and matrix flaws. However, nacre is supposed to be defect-tolerant (Huang and Li, 2013). Many researchers have investigated the reasons for the flaw tolerance of nacre (Barthelat et al., 2007; Wang et al., 2001; Huang and Li, 2013). Though this is the case, studies about the influence of existing defects on the mechanical properties are not available in open literature, especially how the various types of initial flaws affect the strength and toughness. There is a need to systematically study the effect of initial flaws in the matrix and platelet on the mechanical properties such as strength and toughness of nacre. The understanding of how existing defects affect failure, would be useful not only for
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
getting insight into the failure of nacre, but also for developing nacre-inspired artificial materials. Discrete lattice models have been proposed for analysis of deformation and fracture of materials. In a discrete lattice model, the continuum is discretised into one-dimensional elements. The random fuse model is a special case of a discrete lattice model where these one dimensional elements are modelled as resistors (de Arcangelis et al., 1985; Kahng et al., 1988). In a random fuse model, displacement and force are replaced by their electrical analogues; voltage and current. In short, a resistor network replaces the continuum. The random fuse model (RFM) is a scalar analogue of an elastic discrete lattice model. The solution of an electrical network (Zienkiewicz, 1971) is very similar to that of an elastic network. The random fuse model provides a simpler way of solving the discretised continuum. This is because of the scalar nature of the equations used in RFM. Moreover, RFM has only one degree of freedom per node per element. However, the corresponding spring and beam elements have higher degrees of freedom (Skjetne et al., 2001). The random fuse model can incorporate disorder much more easily than finite element models (Alava et al., 2006). RFM has been extensively used to simulate the mechanical behaviour of materials, especially when randomness in the properties and structure is to be introduced in the model (Hansen, 2005; Sahimi, 2003; Herrmann and Roux, 1990). In order to account for evolving failure of materials, a continuous damage random fuse model (CDRFM) has also been proposed (Zapperi et al., 1997). CDRFM has been employed to model domain unfolding in failure simulation of biological composites (Nukala and Šmunović, 2005; Anup et al., 2008). However, in these models, the effect of initial flaws has not been considered. In this paper, we develop a 2D CDRFM model taking into account initial flaws. Initial flaws of the platelet and the matrix are introduced separately into the model. Numerical solution of the model gives the stress–strain response of the composite. Mechanical properties are derived from this response and the effect of initial flaws on these mechanical properties are examined.
Lm
Gap
169
Wp
Mineral Platelet
Organic Matrix
Fig. 1 – Schematic diagram showing the structure of nacre. Note the staggered arrangement of the mineral platelets in the organic matrix.
Shear Matrix Elements
Platelet Elements
Tensile Matrix Elements
Fig. 2 – A part of the CDRFM model showing the various types of elements. Tension elements are used to represent mineral platelets. Shear elements represent matrix connecting the vertical faces of the platelet; tensile matrix elements connect horizontal faces of the platelets.
edges of platelets in the gap region. However, stresses carried by the matrix connecting these horizontal edges of platelets are negligible (Ji and Gao, 2004; Jäger and Fratzl, 2000; Nukala and Šmunović, 2005). Therefore, these tensile matrix ele-
2.
Formulation of the problem
In order to develop a model for failure of nacre, CDRFM is employed. Fig. 1 shows a schematic structure of nacre with the mineral platelets and organic matrix. The model used for simulation is based on this structure and is similar to that employed by other researchers (Nukala and Šmunović, 2005; Anup et al., 2008). A square lattice network of L L size is used. A part of the network used is shown in Fig. 2. In CDRFM, a current–voltage analogue of the elastic model is employed (Zapperi et al., 1997; Nukala and Simunovic, 2005). Ji and Gao (2004) have developed a tension-shear chain model to explain the stress transfer in nacre-like composites. In this model, the mineral platelets are assumed to carry only tensile forces. The matrix is assumed to transfer shear forces between platelets. Based on this model, we assume that platelet elements carry tensile stresses and matrix elements carry shear stresses. These elements are shown in Fig. 2. Tensile matrix elements could be used to connect the horizontal
ments are given zero stiffness values. In this model, all dimensions are normalised with respect to the width of the platelet, Wp (see Fig. 1). A unit width is assumed in the out of plane direction. Therefore, the area of the platelet, Ap is equal to unity. The platelet is divided into a number of elements equal to the normalised length of the platelet, so that the normalised length of each platelet element is unity. A matrix element has a length equal to Lm as shown in Fig. 1. The width of the matrix element is equal to the length of the matrix element. Therefore, the crosssectional area of a matrix element is also equal to one. Stiffness (conductance) of each element is given by Ce ¼ Ae Ee =Le , where Ee is Young's Modulus, Ae is the area, and Le is the length of the element. The stiffness of a platelet element is equal to Young's modulus, Ep since both area and length are unity. The stiffness of a matrix element is found out to be Cm ¼ Gm =Lm , where Lm is the normalised length of a matrix element and G is the shear modulus of the matrix (Nukala and Šmunović, 2005).
170
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
The platelet elements are assumed to be linear elastic. The simulation results of failure of nacre (Nukala and Simunovic, 2005) indicate that allowing platelets to fail, results in a very
We assume a random distribution of initial flaws. It is also assumed that the size of each individual initial flaw is that of a single element of platelet or matrix.
small toughness compared to that of the case without platelet breakage. Such a small toughness is contrary to experimental evidence. Therefore, similar to these studies
3.
Numerical simulations
(Nukala and Simunovic, 2005), platelets are not allowed to fail in this study. The protein, being a bio-polymer, has a sawtooth stress–strain response (Smith et al., 1999). This beha-
A system size of 256 256 is employed for the simulation. The mechanical properties of the platelet and matrix given as an input
viour is due to domain unfolding and is incorporated in the 200
the matrix is assumed to evolve with failure. Therefore, this
180
response is represented by shear fuses with conductance
160
decreasing in discrete steps. When the current exceeds the
140
threshold value (strength) of the matrix elements, matrix element is assumed to fail (domain unfolding), and is termed matrix breakage. In order to model this domain unfolding, the stiffness of the matrix element is reduced by multiplying
Stress (MPa)
model using CDRFM (Nukala and Šmunović, 2005). In CDRFM,
120 100
by a factor of a, where 0oao1. Thus, the conductance of the
60
i
matrix element can be calculated to be Cm a , where i is the
40
number of times it has failed. However, when the maximum
20
shear strain of the matrix is reached, the failure of the
0
molecular backbone (matrix cracking) occurs and the con-
0
0.5
1
1.5
2
2.5
3
3.5 −3
x 10
Strain 200 180 160 140 Stress (MPa)
ductance of the element is made zero. Along the left and right edges of the model, periodic boundary conditions are imposed. A constant voltage difference is assumed between the top and bottom of the model. Matlab package (MathWorks, 2013) is used to solve Kirchoff's equations. Numerically, a unit voltage difference is applied between the top and bottom of the model. The ratio (r) of current in each resistor to the threshold value of the resistor is found out. The element having a maximum value of this ratio fails, and consequently, the conductance matrix is modified. The equation being linear, the external current and voltage are scaled by 1=r, so that the current in the element having maximum r reaches the threshold value. In effect, the applied voltage is increased or decreased so that one fuse alone is broken. The modified conductance matrix is used in the next step for the solution of the modified system. This process of breaking of bonds continues until the lattice separates in two parts. The system current ðIc Þ and the system voltage ðVc Þ of the whole composite are found out each time the equations are solved. The ratio of Ic =L to Vc =L gives the stress–strain behaviour of the composite (Nukala and Simunovic, 2005). The maximum value of stress is considered to be the strength of the composite. The toughness is calculated as the area under the stress–strain curve.
80
120 100 80 60 40 20 0
0
0.5
1
1.5
2
2.5
3
3.5 −3
x 10
Strain
Fig. 3 – Comparison of stress–strain responses for varying values of initial (a) platelet flaws and (b) matrix flaws. The curves shown are for initial flaws of 0% (top-blue), 1%, 2%, 5%, 10%, 15%, and 20% (bottom-dark brown). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
Table 1 – Dimensions of nacre at micro-scale: lengths in nm, area in nm2 (Nukala and Simunovic, 2005; Song et al., 2003). Dimension
Value
Platelet width ðW p Þ
500
Gap Platelet length ðLm Þ Matrix area Matrix length
500 3500 500 25
Normalised value
Number of elements
1
1
1 7 1 1/20
1 7 1 1
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
for simulation are similar to that of the constituents of nacre. Thus, the platelet Young's modulus is assigned a value of 100 GPa (Jackson et al., 1990) and shear modulus of matrix is given a value similar to that of the organic matrix, 4.6 GPa (Jackson et al., 1990, 1988). A typical value of 46 MPa (Jackson et al., 1988; Nukala and Simunovic, 2005) is given for the shear strength of the organic matrix, τs. The maximum shear strain that the protein in nacre can undergo is 200% (Wang et al., 2001). In the present work, a value of 200% is also taken to be the maximum shear strain, ϵmax that the matrix is able to undergo. Each time a matrix element breaks, its stiffness values are decreased to 90% of the original value (a ¼ 0:9Þ in order to simulate the breakage of bonds. The maximum number of breakages for each matrix element, nmax is calculated based on the maximum strain and decrease in stiffness (Nukala and Simunovic, 2005), and is given by nmax ¼ ðlogðϵ0 Þ= ðϵmax ÞÞ=logðaÞ 50, where ϵ0 is found from the relation, τs ¼ ϵ0 Gm . Table 1 gives the dimensions of the model used in the present study. These dimensions are similar to that of the structure of nacre (Nukala and Simunovic, 2005). Table 1 also shows the number of elements used for a platelet and matrix. The effect of initial flaws in the platelet and matrix separately on the stress–strain response is investigated in the present study. The sites of the initial flaws is taken to be at random among all the respective elements. The elements existing at these sites are removed before the start of the simulation. We define, percentage initial platelet flaws, DiP ¼ RP =TP 100, where RP is the number of initially removed platelet elements and TP is the total number of platelet elements in the model. The percentage of initial matrix flaws is also defined in a similar way. A parametric study is conducted by varying the percentage of initial flaws from 0 to 20%. Fig. 3(a) shows the stress–strain response for the case of initial platelet flaws; (b) shows the response in the case of initial matrix flaws.
4.
Discussions
There is a change in the stress–strain response when the percentage of initial flaws is varied as shown in Fig. 3(a) and (b). The stiffness, strength, and toughness of the composite are extracted from the stress–strain response. The overall mechanical properties obtained should not depend on system size, and the system size should be large to avoid any effects due to random variations in locations of flaws. Here, we use a large system size of 256 256. Also, in order to obtain statistically correct values, a number of five runs are made for each value of initial flaws. The results thus obtained for the case without flaws are compared between system sizes of 128 128 and 256 256. The stiffness values of the composite obtained for the two system sizes are 73.52 and 73.48 MPa, and the value of strength for the two system sizes are 182.9 and 182.9 MPa. Thus, the difference in these mechanical properties is very small when the system size is doubled from 128 to 256. Therefore, a system size of 128 128 could even be used. However, we report values for a system size of 256 256, for better accuracy. The tensile modulus and strength of nacre for the case without flaws is compared with the works of Kotha et al. (2001) and Barthelat and Rabiei (2011). The analytical expressions for stiffness and strength of the composite from those
research works are given in the following equations: ! 1 1 1 2 ð1 þ coshðγρÞÞ ¼ 1þ þ Ecomp h Ep ρð1=hÞ Gγm sinhðγρÞ σ comp ¼ ρτs =2
171
ð1Þ ð2Þ
Here, Ecomp and σ comp are the stiffness and strength of the composite, ρ is the aspect ratio of the platelet and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ ½Gm =Ep Þð1=hÞ. Taking ρ ¼ 7, Ep ¼100 MPa and mineral volume fraction ϕ ¼ 20=21, gives the composite strength as 161 MPa and stiffness, Ecomp ¼82.1 GPa. In comparison, the results of the simulation give values of 182.9 MPa and 73.48 GPa. The values are comparable, but the difference could be due to the fact that in the derivation of the analytical expressions, matrix shear stress is assumed to be a constant. In contrast, these stresses are allowed to vary in RFM. Note that for the case without flaws, the results obtained are same as that of Nukala and Simunovic (2005). The stiffness (Young's modulus) and strength obtained agree with the experimental results as well (Jackson et al., 1988). The variation of the mechanical properties with initial flaws is shown in Fig. 4(a)–(c). The mechanical properties of the composite are normalised with respect to the case without any flaws. All mechanical properties decrease with an increase in both types of initial flaws, though at varying rates. The stiffness (defined as the initial linear part of the stress– strain response) decreases at about a constant rate. The strength and toughness decreases rapidly initially, but this rate of decrease becomes lesser with an increase in the percentage of initial flaws. For all mechanical properties, it is seen that platelet flaws are the most detrimental. Matrix flaws have a lesser effect on mechanical properties. Fig. 5(a) and (b) compares how different mechanical properties change as a function of initial platelet flaws and initial matrix flaws. The graphs of mechanical properties shown here are the same as in Fig. 4(c), but are grouped in terms of the type of flaw. It is seen from Fig. 5(a) and (b) that for all types of initial flaws, stiffness has the least change, followed by strength. Toughness shows the highest variation with an increase in initial flaws. Fig. 5(a) and (b) also shows the number of domains unfolded (matrix breakage) for different percentages of initial flaws. These figures also show that when the number of broken matrix elements decreases, the strength and toughness also decrease. The number of broken matrix elements decreases with both types of initial flaws. For the same percentage of initial flaws, initial matrix flaws produce a larger number of broken elements than initial platelet flaws. The trends of decrease in toughness and the number of broken matrix elements are similar. The number of breakages decreases as flaws increase, thereby reducing mechanical properties such as toughness. In the simulation, the failure path consists of fully failed (cracked) matrix elements. Cracked matrix elements denote those elements that have reached their maximum strain and have their conductance set equal to zero. We examine how the failure paths vary with respect to the type and number of flaws. The tortuousness of the failure path is defined by the number of failed elements. Let Ni denote the number of failed (cracked) matrix elements. Fig. 6(a) shows a part of the failure path when no initial flaws are present. The failure path is similar to that
172
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
1 1
Initial Platelet Flaws Initial Matrix Flaws
0.9 0.8
0.9
Normalised Strength
Normalised Stiffness
0.95
0.85 0.8 0.75 Initial Matrix Flaws Initial Platelet Flaws
0.7
0.7 0.6 0.5 0.4 0.3 0.2
0.65
0.1 0
5
10
15
0
20
0
5
10
15
20
Initial Flaws (%)
Initial Flaws (%)
1 0.9
Initial Matrix Flaws Initial Platelet Flaws
Normalised Toughness
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
5
10
15
20
Initial Flaws (%)
Fig. 4 – Comparison of (a) stiffness (b) strength and (c) toughness for different percentages of flaws. These mechanical properties decrease with an increase in the percentage of initial platelet and matrix flaws.
obtained by Nukala and Simunovic (2005). Fig. 6(b) shows the number of failed elements for both types of initial flaw: platelet and matrix. The failure path is tortuous the most when no initial flaws are present (Ni ¼ 1048). The path tortuousness in descending order is for the cases of initial matrix and initial platelet flaws. Path tortuousness decreases with an increase in percentage of the initial flaws. Therefore, the decrease in toughness for both types of initial flaws (platelet and matrix) could be associated with the lessening of path tortuousness. Several experiments were conducted by different research groups to find out the mechanical properties of nacre (Jackson et al., 1988; Bekah et al., 2012). However, the values reported show a wide variation. The largest variation is for toughness, followed by that of tensile strength and the least for stiffness. For example Jackson et al. (1988) gives values of 350–1240 J/m2 for toughness, 70–140 MPa for tensile strength and 60–70 GPa for stiffness. Another research group reported values of 100–1500 J/m2 for toughness, 70–100 MPa for tensile strength and 70–80 GPa for stiffness (Bekah et al., 2012). The results of the current study show a similar trend in the variation of mechanical properties; toughness the largest variation, stiffness the least and tensile strength in between, for both types of initial flaws (for example, see Fig. 5(a)).
Therefore, this variation in mechanical properties reported in literature could be due to naturally occurring defects in nacre. In the simulations reported in Section 3, the platelets were assumed not to fail. Here, this assumption is critically examined. The maximum stress in the platelets, over the whole simulation, σs for case without flaws is found to be 366 MPa. Since, the platelets in nacre have a strength E400 MPa, the assumption seems to be correct. Researchers have used analytical models to predict the relationship between the tensile stress in platelets and shear stress in matrix, σ ¼ ρ τ (Ji and Gao, 2004), where ρ is the aspect ratio of the platelet. This gives a value of σ s ¼ 7 46 ¼ 322 MPa, which is comparable to that of the present study. The difference in the values could be due to the assumption of constant shear stress made in the analytical model. In contrast, the RFM distributes stress in the matrix to attain equilibrium, and in the process permits variation in shear stress to be developed. Further, we examine the effect of initial flaws on the maximum tensile stress in the platelet over the whole simulation. Fig. 7 shows the variation of this stress with the percentage of flaws. The value of maximum tensile stress decreases with an increase in initial platelet and matrix flaws. This means that if platelet breakage does not happen for the case without flaws, this will not occur for the case
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
Normalised mechanical properties/ Normalised no. of broken elements
1.2
Normalised Stiffness Normalised Strength Normalised Toughness Normalised No. of Broken Elements
1
0.8
0.6
0.4
0.2
0
0
5
10 Initial Platelet Flaws (%)
15
20
1
Normalised mechanical properties/ Normalised no. of broken elements
0.9 0.8 0.7 0.6 0.5 0.4
Normalised Stiffness Normalised Strength Normalised Toughness Normalised No. of Broken Elements
0.3 0.2 0.1 0
0
5
10
15
20
Initial Matrix Flaws (%)
Fig. 5 – Comparison of normalised mechanical properties and normalised number of broken elements for various percentages of initial (a) platelet flaws and (b) matrix flaws. The curves shown are the same as in Fig. 4(a)–(c), but are grouped in terms of the type of flaws.
with initial platelet or matrix flaws. This fact could be made use of in the design of bio-inspired composites. However, note that even with fully matrix failure, these composites with initial flaws has lesser toughness and strength than that of the case where no flaws are present. The above discussion shows that flaws influence the stresses in the elements, and thereby changes the way the elements break. In order to understand how initial flaws influence the stress distribution in the model, we compare the stresses developed in all the elements in the entire model, before failure of any of the elements occurs. Fig. 8(a) shows the stress distribution for the case without any flaws, and (b) and (c) show the stresses when matrix and platelet flaws of 20% are present. Here, the subscripts 0, 1 and 2 denote the following cases: no initial flaws, initial matrix, and initial platelet flaws respectively. All models (without flaws, and with initial matrix and platelet flaws) are subjected to the same displacement, and stresses developed in the elements are found out using RFM. The stresses are normalised with respect to the maximum shear stress in the model for the case without flaws. From 8(a) it can be seen that the stresses in the model are in some specific values when there are no flaws. In contrast, stresses are distributed over a range of values when flaws are present (see Fig. 8(b) and (c)). For the case without flaws, ratio of the maximum tensile stresses in the platelet to that of the maximum shear stress in the matrix, R 5. However, when there is initial flaws (20%) in the matrix, the matrix shear stresses increase by a larger amount than that of the platelet tensile stresses, making this ratio R 2:2, as seen in Fig. 8(b). A similar pattern of increase in the stresses in the platelet and matrix is observed for the case of initial flaws (20%) in the platelet, and R 1:7 (refer Fig. 8(c)) in this case. Therefore, due to initial flaws, though there is an increase of the tensile stress in the platelets, there is a higher increase of shear stress in the matrix. Considering the whole simulation, since the matrix failure occurs at the shear strength of the matrix, this stress distribu-
1100
No. of Cracked Elements
1000
Initial Matrix Flaws Initial Platelet Flaws
900 800 700 600 500 400 300 200
0
5
10
15
20
Initial Flaws (%)
Fig. 6 – (a) Model showing a part of the failure path (cracked matrix elements) in the case of no initial flaws and (b) variation of number of cracked matrix elements with the percentage of initial flaws. The number of cracked matrix elements decreases with an increase in initial flaws.
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journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
tion means that matrix failure becomes increasingly likely for the case with flaws, when compared to that without. Thus, this matrix failure prevents higher stresses being developed in the
platelet. Consequently, maximum stress in the platelets over the whole simulation is not increased due to flaws. However, note that the stresses shown in Fig. 8(a)–(c) are before breakage of any element. As the simulation proceeds, matrix elements break, changing the stress in all the elements after each break. In order to develop bio-inspired materials, following could be the suggestions based on the results of this work
1 0.98
Normalised Stress
0.96 0.94 0.92
1. Bio-inspired composites could obtain better mechanical properties than that of nacre by eliminating or reducing initial flaws. 2. Priority should be given to reduce initial platelet flaws than matrix flaws to get highest improvement in mechanical properties.
Initial Platelet Flaws Initial Matrix Flaws
0.9 0.88 0.86 0.84 0.82 0
5
10
15
In the present study, we assume that matrix at the end of the platelets could not transfer any load. However, some studies indicate that the matrix connecting these short platelet edges could influence the stress distribution in platelets (Bekah et al., 2012). The influence of such tensile elements will be incorporated in future studies. We used the random fuse model in this work to find the effect of flaws on the overall mechanical
20
Initial Flaws (%)
Fig. 7 – Variation of maximum tensile stress in the platelet, for the case with initial flaws. Stress is normalised with respect to that of the case without flaws. Maximum tensile stress decreases with an increase in the number of flaws.
7
5
6
4
τ0 / τ0max
3.5
σ 0 / τ0max
τ1 / τ0max σ 1 / τ0max
5 Normalised Stress
Normalised Stress
4.5
3 2.5 2 1.5
4 3 2
1
1 0.5 0
0
2
4
6
8
10
12
0
14
0
2
4
6
8
4
10
12
14 4
x 10
Element Number
Element Number
x 10
10 9
τ /τ 2
8
2
Normalised Stress
0max
σ /τ
0max
7 6 5 4 3 2 1 0
0
2
4
6
8
10
12
14 4
Element Number
x 10
Fig. 8 – Comparison of the stresses in the model before any breakage of elements, when subjected to the same displacement at the boundary: (a) no initial flaws; (b) initial matrix flaws (20%); (c) initial platelet flaws (20%).
journal of the mechanical behavior of biomedical materials 46 (2015) 168 –175
properties. However, other tools such as finite element methods (FEM) do have certain advantages over the RFM. In FEM, an extensive analysis of the stress distribution in the platelet and matrix near the sites of flaws could be undertaken. These analyses would form part of the future study.
5.
Conclusions
A continuous damage random fuse model for nacre was developed which takes into account the effect of initial flaws in the platelet and the matrix on the composite mechanical properties. The following are the major conclusions.
All mechanical properties show a decrease with an
increase in initial flaws, though at varying rates. Stiffness decreases at about a constant rate. All other mechanical properties show a rapid decrease for lower values of initial flaws, but with further increase in initial flaws, rate of decrease reduces. The mechanical properties most influenced by initial flaws in descending order are toughness, strength, and stiffness. Decrease in toughness due to the presence of initial damage could be attributed to the decrease in the number of unfolded domains and reduction in tortuousness of the failure path. Variation in the reported experimental results of mechanical properties could be due to initial flaws present in nacre. Initial flaws cause higher increase in shear stress in the matrix than in the platelet.
These results could be important in understanding how nacre achieves its mechanical properties, especially toughness. The information obtained would also be useful in the design of bio-inspired composites.
Acknowledgement The author would like to thank Dr. Arun C.O. and Dr. Praveen Krishna I.R. of IIST for useful discussions and suggestions.
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