Effect of Size and Volume on the Breaking Properties of Fragile Materials: The Case of Laminate for Orthopedic Acrylic Glass-Perlon Use

The Weibull model is a probabilistic model of failure behavior obeying the weakest link theory "Weakest Link Theory W.L.K". This model is based on the fact that the resistance of a chain is governed by that of its weakest link. It is the same for a set of chains. Applying this model to a fragile material makes each link correspond to a volume element, the weakest links corresponding to the volume elements containing the most critical defects.

Let $\left(\Delta P_{f}\right)_{i}$ be the probability of failure of an element i of a structure. His probability of survival.

${{\left( {{P}_{s}} \right)}_{i}}=1-{{\left( \Delta P{}_{f} \right)}_{i}}$      (1)

The probability of failure Ps of the structure is the product of the probability of survival of each element:

${{P}_{s}}={{\pi }_{i}}{{\left( {{P}_{s}} \right)}_{i}}={{\pi }_{i}}\left\{ 1-{{\left( {{P}_{f}} \right)}_{i}} \right\}$      (2)

with $\left(\Delta P_{f}\right)_{i}$ small, we can write:

${{P}_{s}}=\pi \exp \left[ -{{\left( \Delta {{P}_{f}} \right)}_{i}} \right]=\exp \left[ -\sum{{{\left( \Delta {{P}_{f}} \right)}_{i}}} \right]=\exp \left( -B \right)$    (3)

B is defined as "the risk of rupture". To evaluate B, Weibull added the probabilities of failure of all the elements of volume $\Delta V$ in the structure. The probability of failure is given by:

${{\left( \Delta {{P}_{f}} \right)}_{i}}=\varphi \left( \sigma  \right)\Delta {{V}_{i}}$      (4)

φ(σ) Represents the probability that a resistance defect of less than σ exists per unit volume.

$B={{\sum{\left( \Delta P{}_{f} \right)}}_{i}}=\sum{\left[ \varphi \left( \sigma  \right).\Delta {{V}_{i}} \right]}=\int\limits_{V}{\varphi \left( \sigma  \right)}dV$       (5)

The probability of rupture of the whole body then becomes:

${{P}_{f}}=1-{{P}_{s}}=1-\exp \left( -B \right)=1-\exp \left[ -\int\limits_{V}{\varphi \left( \sigma  \right)}dV \right]$      (6)

In the case of simple traction which is the case considered by Weibull. The Function φ(σ) which expresses the resistance properties of the material is given by:

$\varphi \left( \sigma  \right)={{\left( \frac{\sigma -{{\sigma }_{u}}}{{{\sigma }_{0}}} \right)}^{m}}$       (7)

σ and σ_u are respectively the applied stress and a threshold stress below which the probability of failure is zero. We often consider σ_u=0.

The exponent m represents the modulus of Weibull, which it's an inhomogeneity parameter which characterizes the extent of the breaking stress distribution and the influence of the size.

σ₀ is a normalization constraint that represents a rupture probability with a value of 0.632 for a unit volume [1].

The settings σ₀ and m are considered properties of the material. Thus the risk of failure only depends on the stress distribution in the test specimen.

For a volume V_t subjected to uniform tension σ_t, the Eq. (6) is written:

${{P}_{f}}\left( {{\sigma }_{t}} \right)=1-\exp \left[ -{{V}_{t}}{{\left( {}^{{{\sigma }_{t}}}/{}_{{{\sigma }_{0}}} \right)}^{m}} \right]$      (8)

which can also be written:

${{P}_{f}}\left( {{\sigma }_{t}} \right)=1-\exp \left[ -{{\left( {}^{{{\sigma }_{t}}}/{}_{{{{\bar{\sigma }}}_{t}}} \right)}^{m}} \right]$     (9)

$\bar{\sigma}_{t}$ represents the normalization stress in tension:

${{\bar{\sigma }}_{t}}={}^{{{\sigma }_{0}}}/{}_{\left( {{V}_{t}} \right)}\tfrac{1}{m}$      (10)

In the case of 3-point bending; Sbiai et al. [4] by integrating Eq. (5), allowed to write:

${{P}_{f}}\left( {{\sigma }_{b}} \right)=1-\exp \left[ -{{\left( {}^{{{\sigma }_{b}}}/{}_{{{{\bar{\sigma }}}_{b}}} \right)}^{m}} \right]$     (11)

$\bar{\sigma}_{t}$ represents the normalisation stress in bending:

${{\bar{\sigma }}_{f}}={{\sigma }_{0}}{{\left[ {}^{2{{\left( m+1 \right)}^{2}}}/{}_{{{V}_{b}}} \right]}^{{\scriptstyle{}^{1}/{}_{m}}}}$      (12)

Thus the 3-point bending test results in a two-parameter Weibull distribution with the same factor m as for the tensile test but a different normalization factor. For equal probabilities of rupture, we can then write:

$\left( {}^{{{\sigma }_{f}}}/{}_{{{\sigma }_{t}}} \right)={{\left[ 2{{\left( m+1 \right)}^{2}}\left( {}^{{{V}_{t}}}/{}_{{{V}_{b}}} \right) \right]}^{{\scriptstyle{}^{1}/{}_{m}}}}$       (13)

The different constraints σ_i-are listed in ascending order of σ₁ to σ_N-and assigned a probability P_f function of rank i. The analytical calculation of P_f is done from one of the expressions of ‘Median Rank M.R’:

${{P}_{f}}=MR1=\frac{i}{N+1}$      (14)

${{P}_{f}}=MR2={}^{\left( i-0.3 \right)}/{}_{\left( N+0.4 \right)}$      (15)

Expression (14) is the most used. The graphic representation of $\operatorname{Ln} . \operatorname{Ln}\left[\frac{1}{\left(1-P_{f}\right)}\right]$-as a function of the logarithm of the applied stress is a line of slope m.

4.1 Examples of load-displacement and stress-strain curves

Figure 5 shows an example of load-displacement curves in direct traction and in 3-point bending of acrylic resin. These curves are linear until they break, reflecting the fragile elastic nature of this polymer in these two types of stress.

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Figure 5. Example of average load-displacement curves

The results of measurement of the stress at break as well as the average values and the coefficients of variation for each stress are collated in Table 1.

Table 1. Measured values of the stress at break [MPa]

	Bending test	Tensile test
Constraint (MPa)	27 (08%)	69 (12%)

The mean flexural stress at break is much higher than that measured in direct traction. On the other hand, the deformations measured on the tensile face at the maximum bending load are greater than those measured in direct tension.

Figure 6 shows the evolution of the load - deflection curves in the direction of molding. The same type of behavior is observed in the perpendicular direction.

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Figure 6. Example of load - deflection curve reading.

The evolution of the stress-strain curve for the P-V-2P-V-P architecture in both molding flow directions is illustrated in Figure 7.

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Figure 7. Variation the "σ" as a function of "ɛ" for P-V-2P-V-P architecture

These curves present the same appearance in general and are initially linear materializing the elastic response of these materials then deviate from the linearity reflecting the damage which takes place and develops within the volume in the form of a tear of the composite before the fall. sudden and unstable load.

Table 2 collates the results of measurement of the mechanical characteristics. Numbers in parentheses indicate standard deviations. The mean flexural stress at break as well as the strain measured on the tensile face are greater than those measured in direct tension.

	Bending test		Tensile test
Direction	Molding direction	Perpendicular direction	Molding direction	Perpendicular direction
Constraint (MPa)	48 (17%)	86 (15%)	26 (8.5%)	49 (9%)
Young's modulus (MPa)	2640 (15%)	3687 (9%)	1307 (6.5%)	1738 (8.5%)

The results obtained are characterized by a dispersion recognized as an intrinsic characteristic of composite materials. The main cause of this dispersion is the structure heterogeneity of materials as well as the existence of several defects such as defects in size, dimension, orientation and different densities within the volume. These defects are induced in various ways either during the elaboration or during preparation of specimens.

The results of the two tests carried out show that in the direction perpendicular to the molding, the stresses values and elasticity moduli are increased compared to the values measured in molding direction. This can be interpreted by the fact that the mode of print run of glass fabric is slightly free in molding direction than in perpendicular direction, where the knots are very tight thus causing a stresses concentration. On the other hand, the presence of friction forces between the glass fibers in knots contribute to amplify the phenomenon.

4.2 Weibull probabilistic model application units and numbers

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Figure 8. Curve indicating for each stress level the failures probability in the case of acrylic resin

The results of the stress at break are analyzed by Weibull's statistical theory. The analytical calculation of the probability of failure was performed from the expression (14) for "median rank". Figure 8 shows the variation in the probability of failure as a function of the tensile strength of the acrylic resin for the two stresses. Indeed, Figure 9 shows for each stress level of studied architecture the evolution of failure probability in both directions of molding flow, where all the curves are sigmoids representing Weibull distributions of breaking stresses.

9.png

Figure 9. Curve indicating for each level of constraint the failures probability in the case of studied architecture

Statistical analysis of these results by Weibull theory leads to very similar Weibull modulus values in these two types of stress as shown in the Figure 10 and 11. The slight difference observed can be attributed to the experimental conditions.

The dispersion of the measurement results is therefore a characteristic of these polymers & composites which are above all subject to a size effect linked to the requested volume and to the stress distribution in this volume. Thus the probability of finding a critical defect in a bar loaded in tension is greater than that in a bending specimen of smaller volume. Because of these effects the stress measured in bending is greater than that measured in tension. Table 3 shows that the ratio σ_rf /σ_rt calculated from the Weibull model is almost identical to that measured by experiment.

10.png

Figure 10. Weibull diagram

It is interesting to note that many studies have attempted to correlate the Weibull modulus with the parameters for measuring dispersion observed on the rupture stresses and especially the variation coefficient.

Broutman and Krock [12] estimate that for a variation coefficient of 9% generally measured on the dispersion of rupture stresses of composite materials, the Weibull modulus is the order of 7.

Zweben and Rosen [13] inversely connects the Weibull modulus "m" and the variation coefficient "c" of tensile stresses of composites by the following relationship:

m = 1.2 / c     (16)

The same relationship was suggested by Margetson and Brokenbrow [14] who reports that composite materials containing a moderate concentration of fibers show a significant variability of rupture stresses characterized by values of "m" between 12.9 and 16.6.

Margetson and Brokenbrow [14] propose the following relationship:

c = 1.27 / (m + 0.56)      (17)

In addition, the measured experimental values of Weibull modulus "m" in both directions for the resin as well as the composite material and those calculated from the empirical relations (16) and (17) are recorded in Table 4.

The values of the Weibull modulus “m” calculated by the formulas (16) and (17) are clearly higher than the experimental values. Thus, this difference is due to the degree of dispersion observed on the measured values of the breaking stresses of the material studied. The sources of this dispersion are various, including defects on the surface layer of the sample as well as volume defects (shapes, sizes, orientations…….). These parameters are of little significance to give an adequate correlation.

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Figure 11. Graphical representation of the linear probability equation for the P-V-2P-V-P architecture

Table 3. Report σ_rf /σ_rt theoretical and experimental

report	Weibull	experimental
σrf / σrt	2.83	2.72
Molding direction σrf / σrt	2.20	1.85
Perpendicular direction σrf / σrt	2.17	1.76

4.3 Mechanisms of rupture and ruin

According to the microscopic examinations, the tear observed follows a plane perpendicular to applied load [15]. The extension of this tear is revealed by a modification in appearance and a discoloration of resin which tends towards a white color as shown in Figure 12.

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Figure 12. 3D microscopic observations of resin fracture facies

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Figure 13. SEM observations of composite fracture facies

Microscopic observations have shown that the tear generally follows the path of the knots of the glass fabric along a plane perpendicular to the axis of load application in the tensile test. The propagation of the rupture is always accompanied by a change in appearance and a discoloration of the composite which turns towards the white color (Figure 13). This is probably the consequence of the first debonding and loosening due to the effect of the applied stress. Some specimens did not show breakage by separation; the two pieces remain attached but show no resistance to the load. The tear generally shows loosening of glass fibers and perlon. These take place without detachment of the matrix suggesting a fiber-matrix interface less resistant than the matrix itself. Other observations made it possible to locate the rupture in the inter-layering spaces as well as in the tissue at the level of the binding points. Interfacial decohesions as well as a brittle fracture of the matrix initiated at the porosities and voids are also observed. Finally, let us point out that the paths of the cracking remain very complex to locate and generally follow tortuous paths.

It should be noted that the deformation and damage mechanisms of woven structures are very complex to identify and analyze because of the complicated structure to model of weaving, binding points, … etc. An interesting study on the characterization of structures fabrics has provided information on tissue damage mechanisms [16].