Accurate Critical Stress Intensity Factor Griffith Crack Theory Measurements by Numerical Techniques

Abstract

Critical stress intensity factor (K_Ic) has been an approximation for fracture toughness using only load-cell measurements. However, artificial man-made cracks several orders of magnitude longer and wider than natural flaws have required a correction factor term (Y) that can be up to about 3 times the recorded experimental value [1-3]. In fact, over 30 years ago a National Academy of Sciences advisory board stated that empirical K_Ic testing was of serious concern and further requested that an accurate bulk fracture toughness method be found [4]. Now that fracture toughness can be calculated accurately by numerical integration from the load/deflection curve as resilience, work of fracture (WOF) and strain energy release (S_Ic) [5, 6], K_Ic appears to be unnecessary. However, the large body of previous K_Ic experimental test results found in the literature offer the opportunity for continued meta analysis with other more practical and accurate fracture toughness results using energy methods and numerical integration. Therefore, K_Ic is derived from the classical Griffith Crack Theory [6] to include S_Ic as a more accurate term for strain energy release rate (𝒢_Ic), along with crack surface energy (γ), crack length (a), modulus (E), applied stress (σ), Y, crack-tip plastic zone defect region (r_p) and yield strength (σ_ys) that can all be determined from load and deflection data. Polymer matrix discontinuous quartz fiber-reinforced composites to accentuate toughness differences were prepared for flexural mechanical testing comprising of 3 mm fibers at different volume percentages from 0-54.0 vol% and at 28.2 vol% with different fiber lengths from 0.0-6.0 mm. Results provided a new correction factor and regression analyses between several numerical integration fracture toughness test methods to support K_Ic results. Further, bulk K_Ic accurate experimental values are compared with empirical test results found in literature. Also, several fracture toughness mechanisms are discussed especially for fiber-reinforced composites.

1. Introduction

1.1 Problems with K_Ic Test Methods and Numerical Answer

K_Ic measurements have been a difficult means to compare materials for a type of fracture toughness related to the material modulus and fracture energy during crack propagation through numerous empirical formulas and standards developed from practice [1, 2]. As such, K_Ic testing proceeds without regard to flexural bending or shear theory from Strength of Materials through Engineering Science [7]. However, K_Ic has nevertheless been applied regularly through committee standards for fracture toughness while routinely discarding valuable deflection data instead using only the most basic load-cell measurements for maximum strength. [1-3].

Subsequent ability to calculate energy adsorbed before critical maximum strength failure as WOF or even energy released as 𝒢_Ic [5, 6] is then lost with current K_Ic test methods. Countless fracture toughness test protocols for K_Ic have been used with different sample sizes, sample dimensions and testing fixtures so that comparative analysis between laboratories is a genuine concern due to differences in results of data, difficulty in the duplication of identical test procedures and even experimental inaccuracy making most K_Ic tests unreliable [8]. Further, attempts to simulate natural cracks for testing have resulted in undesirable man-made notches or machined defects with artificial razor cuts that have become standard practice applied in K_Ic flexural test methods [1-3]. Also, in K_Ic beam testing artificial defects often extend into and even beyond the neutral plane [1-3]. Defects into the neutral plane area thus remove most tensile forces [7]. In addition, loading flexural K_Ic testing samples in shorter beam fixtures place materials into shear compressive loads so that the equations for flexural strength or modulus cannot be calculated with accuracy suggested by Mechanics of Materials Science [7]. Since K_Ic is considered to be sensitive to local conditions even as microscopic voids/porosity [4, 8], machined defects on the order of 1.0 mm and larger that can change between test protocols and even within the same test procedure [1-3] would be considered of major concern for error. In fact, problems with K_Ic testing were serious enough in 1980 for the National Academy of Sciences (NAS) to request that limitations be set on K_Ic theory [4]. Further, the NAS was interested in solutions for inter-laboratory agreement to provide bulk fracture toughness testing that reproduces only consequences from natural flaws rather than man-made defects and compares accurately to tensile strength data results which have provided the majority of knowledge on the fracture of brittle ceramics [4]. However, since strength is an indication of inherent fracture toughness, military specifications MIL STD 1942(MR) for brittle materials were adapted soon thereafter in 1983 to provide reliable quality control ceramic testing using four-point flexural fixtures for tensile failure from the results on strength and modulus [8].

Currently a simple numerical solution by integration with computer spreadsheet technology provides an answer for easy and accurate bulk fracture toughness methods [5, 6]. With highly accurate load-deflection mechanical test machines available in widespread use, correct bulk material properties can be measured with adequate flexural test spans to prevent shear error for true fracture toughness by energy methods as resilience, work of fracture and critical tensile mode I strain energy release (S_Ic now for better application in formulas and presented previously as SER_Ic) [5, 6]. However, the use of the many K_Ic methods [1, 8] has produced an extensive body of information that might be available for literature meta analysis in the future even though bulk material energy-method testing can be easily applied. Using numerical integration techniques [5, 6], the study of crack propagation can be investigated by a micromechanics fracture toughness analysis in relation to specimen deflection and the energy beyond maximum peak load where unstable damage initially occurs as strain energy release or again S_Ic as only a small fraction of the total energy adsorbed during fracture [5, 6], Figure 1. Toughness values for SER_c or S_c are similar to strain energy release rate (𝒢_c) where c represents the maximum or critical load [5, 6]. However, further be clear that 𝒢_c is not a time dependent rate but rather a toughness value expressed in J/m² and was originally derived for the change in energy consumed during crack propagation at critical load with respect to the increasing surface area on one side of the crack for a definition that is not practically applied. S_c on the other hand is basically applied from the load-deflection curve as the total energy released per sample area for width and thickness from maximum or critical load to complete failure [5, 6].

Figure 1.

Example of a typical load deflection curve from a particulate-filled composite compared to a fiber-reinforced composite that accentuates S_Ic with sample toughness from fiber reinforcement to more easily appreciate the energy released during crack propagation at failure. With permission: Polymer Composites Volume 28: 311-324. 2007

1.2 Theoretical Griffith Fracture Toughness

Modulus (E) and 𝒢_c have been derived previously as the square root product (E𝒢_c)^1/2 for the critical stress intensity factor (K_c) [1, 2, 9-12]. In 1920, Griffith showed that a crack would propagate when the strain energy per unit of crack surface energy (γ) exceeds the material atomic bond energies to generate two new surfaces [9] by the formula:

$$\mathrm{\sigma }={(2\mathrm{E}\mathrm{\gamma }/\mathrm{\pi }\mathrm{a})}^{1/2}$$ [1]

Where σ is the applied stress, E the material modulus, and “a” is the length of a surface crack or half the crack length of an internal crack. Thermodynamics were further considered for the energy of both fracture surfaces [1, 10-14] such that the rate of toughness lost definition as 𝒢_c was redefined through a conventional toughness value as cross-sectional area of the sample at the crack and reinserted for γ:

$${\mathrm{𝒢}}_{\mathrm{c}}=2\mathrm{\gamma }$$ [2]

Since 𝒢_c is the crack force per crack area beyond a critical value, 𝒢_c actually represents the work of fracture or fracture energy of the propagating crack from maximum load to complete failure [1, 5, 6, 10-14] so equation 1 becomes:

$$\mathrm{\sigma }={(\mathrm{E}{\mathrm{𝒢}}_{\mathrm{c}}/\mathrm{\pi }\mathrm{a})}^{1/2}={(\mathrm{E}\ast {\text{SER}}_{\mathrm{c}}/\mathrm{\pi }\mathrm{a})}^{1/2}={({\text{ES}}_{\mathrm{c}}/\mathrm{\pi }\mathrm{a})}^{1/2}$$ [3]

Where S_c now represents the energy released per total material cross-sectional area at the fracture [5, 6]. The S_c can be determined directly by numerical integration for energy from the maximum load to a predetermined magnitude deflection which is normalized for toughness by the cross sectional fracture surface area with isotropic brittle materials that have been fractured through [5, 6]. On the other hand, fiber-reinforced composites do not generally exhibit complete fracture through the material due to lateral crack deflections [5, 6, 15]. However, the surface area can be derived as the cross sectional area at the site where the crack initiates. By rearranging equation 3:

$$\mathrm{\sigma }{(\mathrm{\pi }\mathrm{a})}^{1/2}={(\mathrm{E}{\mathrm{𝒢}}_{\mathrm{c}})}^{1/2}={({\text{ES}}_{\mathrm{c}})}^{1/2}$$ [4]

Also, the stress intensity factor in mode I tension (K_I) reaches a critical (c) value such that fracture occurs when K_I=K_Ic [1, 2, 4, 6]. Because K_Ic is derived as (E𝒢_Ic)^1/2 [1, 4, 10-14] then

$${\mathrm{K}}_{\text{Ic}}=\mathrm{\sigma }{(\mathrm{\pi }\mathrm{a})}^{1/2}={(\mathrm{E}{\mathrm{𝒢}}_{\text{Ic}})}^{1/2}={({\text{ES}}_{\text{Ic}})}^{1/2}$$ [5]

Since the critical fracture toughness parameter such as K_Ic estimates material properties reflected by crack lengths under specific loading conditions within a material, a geometrical correction factor Y has been approximated to be 1.1215 for a simple edge crack as a boundary condition for initial failure during flexural testing [1, 2]:

$${\mathrm{K}}_{\text{Ic}}=\mathrm{Y}\mathrm{\sigma }{\left(\mathrm{\pi }\mathrm{a}\right)}^{1/2}=\mathrm{Y}{\left(\mathrm{E}{\mathrm{𝒢}}_{\text{Ic}}\right)}^{1/2}=1.1215{\mathrm{(}\mathrm{E}{\mathrm{𝒢}}_{\text{Ic}})}^{1/2}=1.1215{{\text{(ES}}_{\text{Ic}})}^{1/2}$$ [6]

Numerous empirical formulas are available for many different crack defects and loading conditions all following the basic formula for K_Ic in mode I tensile failure [1, 2, 8]. By using energy-type analysis methods, the value for a, as the starter crack length, is not required for calculations so that E and S_Ic can be determined directly with data from the load-deflection curve thereby providing an estimator of crack-type fracture failure [5, 6]. The load-deflection K_Ic therefore becomes a measure of a critical stress at a propagating crack within a material stressed sufficiently to create the initial maximum allowable crack length as a final defect for failure at maximum load during mechanical testing [1, 2, 4, 10-14].

1.3 Bulk K_Ic Measurements by Numerical Integration

Empirical formulas from experimental results combined with the Griffith-fracture theory provide many practical methods for obtaining predictors of material fracture [1, 2, 10-14]. A more generalized accurate means is also available using numerical integration to supply test data for correlations with basic fracture mechanics theory to establish predictors regarding crack propagation [5, 6]. Therefore, a working model for the critical stress intensity factor (K_Ic) by Equation 6 (in units of MPa*m^1/2 as N*m^1/2/area where Pa = N/m² and area is in m²) can be considered by calculating the square root of the combined product between modulus (GPa as force/area) and S_Ic or 𝒢_Ic (kJ/m² as force*m/area). Similar K_Ic MPa*m^1/2 units can be shown to be the square root of the product between strength (MPa as force/area) and WOF (kJ/m² as force*m/area) wherein the stress condition surrounding the propagating microcracks are considered in terms of the plastic deformation leading to failure at maximum loading. It is therefore possible that more accurate correction factors may be obtained directly through calculations of energy from load-deflection results by including a second K_Ic as a correction term based on strength and WOF results. This second K_Ic could better account for geometrical arrangements that combine as a part of the formation of the starter crack, a. K_Ic would then reflect multiple mechanical properties from accurate mechanical test machine data for maximum strength, modulus, WOF and S_Ic for a single fracture predictor that may prove especially worthwhile with high-end brittle materials.

For a brief analysis of a, the theoretical starter crack length at maximum strength can be evaluated from the experimental data and with Equation 5 by rearranging as:

$$\mathrm{a}={\text{ES}}_{\text{Ic}}/\mathrm{\pi }{\mathrm{\sigma }}^2={({\mathrm{K}}_{\text{Ic}})}^2/\mathrm{\pi }{\mathrm{\sigma }}^2$$ [7]

2. Experimentation

2.1 Materials

Quartz fibers at 3.0 mm length (Saint Gobain, QPC Products, Lexington, KY), Table 1, silanated with 3-methacryloxyproplytrimethoxysilane (MPTMS) (DOW Chemical, Midland, MI) were impregnated with a photocure vinyl ester resin system previously described elsewhere [5, 6, 15]. The phototcure vinyl ester resin preimpregnated fibers were then mixed in combination with a zirconia-silicate filler (Z100 by 3M Corporation, St. Paul, MN) to make a photocure paste for composites with fibers from 0.0 wt % to 70.0 wt % for final fiber volume percentages (V_fs) of 0.0 V_f, 5.4 V_f, 10.3 V_f, 19.8 V_f, 28.2 V_f, 35.8 V_f, 42.8 V_f and 54.0 V_f [6, 15]. Quartz fibers with lengths of 0.5 mm, 1.0 mm, 2.0 mm, 3.0 mm and 6.0 mm were preimpregnated with the photocure resin system [5]. The resultant paste with additional zirconia silicate in a photocure vinyl ester resin compound was then used to incorporate all fiber length groups preimpregnated with photocure resin for final moulding compounds of 30 wt % fibers or also 28.2 V_f [5].

Table 1. Fiber Volume Percent Mechanical Test Results-3 mm Fibers (StDev).

Fiber Volume Percent	Flexural Strength (MPa)	Modulus (GPa)	Strain (10-2)	Yield Strength (MPa)	Resilience (kJ/m2)	WOF (kJ/m2)	SIc (kJ/m2)	KIc (MPa*m1/2)	a (um)
0.0 (Z100)	117.56 (5.5)	19.5 (1.3)	.79 (.18)	95.4 (14.6)	3.03 (.99)	4.48 (.42)	0.036 (.024)	1.71 (.30)	68.8 (23.5)
5.4	135.53 (6.8)	22.5 (1.7)	.76 (.16)	109.2 (9.5)	3.35 (0.66)	6.16 (0.36)	0.439 (.133)	4.56 (0.34)	361.0 (31.5)
10.3	204.8 (28.4)	23.4 (0.9)	1.07 (.22)	150.2 (14.2)	6.26 (0.98)	14.33 (2.86)	0.799 (0.670)	6.50 (2.18)	328.5 (176.5)
19.8	259.3 (40.8)	22.7 (2.0)	1.22 (.15)	207.0 (38.9)	12.19 (4.03)	20.02 (5.42)	0.818 (0.509)	7.32 (1.74)	271.8 (121.8)
28.2	374.9 (11.9)	31.5 (1.2)	1.31 (.04)	343.5 (37.1)	23.31 (5.35)	30.11 (1.51)	2.400 (.534)	13.51 (1.15)	415.6 (67.1)
35.8	348.7 (55.3)	35.2 (3.5)	1.25 (.14)	332.8 (37.3)	19.46 (3.98)	27.14 (7.72)	2.327 (0.177)	11.07 (4.49)	410.6 (250.8)
42.8	398.1 (62.8)	39.0 (2.4)	0.0125 (0.0015)	362.1 (76.5)	21.31 (8.00)	28.54 (7.19)	1.920 (0.531)	13.46 (2.03)	368.0 (57.4)
54.0	421.3 (35.3)	42.0 (1.6)	0.0118 (0.0009)	348.0 (64.6)	18.17 (1.80)	27.70 (6.57)	2.200 (0.915)	14.45 (2.89)	392.2 (148.1)

For a better K_Ic analysis comparison with the anisotropic fiber-reinforced composites, other basically isotropic materials were prepared for mechanical testing. A commercial photocure composite, ALERT®, (Jeneric Pentron, CT) was tested for comparison with microfibers having lengths approximately 40 micrometers with 10-micrometer diameters that in turn cannot provide critical lengths for fiber-reinforcement [5, 6, 15]. The Alert® fiber volume fraction is estimated by the manufacturer at approximately 26 V_f. Further, more isotropic materials were tested having equal or uniform properties in all directions similar to the Z100 0.0V_f sample group. A chemical cure acrylic bone cement was included (Codman's Cranioplasty (Johnson and Johnson Codman & Shurtleff, Inc., DePuy International, Raynham, MA). Further, several other isotropic materials were received as small manufacturer specimens with Paradigm particulate-filled polymer CAD/CAM blank (3M Corporation, St. Paul, MN), alumina ceramic Al-300 (Coors Ceramics, Goldin, Co), and tungsten carbide cermet 22 % cobalt binder (Basic Carbide, Buena Vista, PA).

2.2 Experimental Methods

2.2.1 Fully articulated flexural test specimen preparation

Samples 2×2×50 mm accommodating a 40 mm test span were prepared with a split mould clamped between two glass plates. An Epilar 3000 (3M Corporation, St. Paul, MN) was used for the photocure initiation and monitored with a Demetron Radiometer daily to ensure intensities of concentrated light at a wavelength of 470 nanometers were above 500mW/cm². Excess material was removed from each sample followed by a sanding process down to 600 silicon carbide grit. Samples were then placed in a 37 degree C water bath for 24 hours, primarily as a control for a uniform postcure before mechanical testing. Smaller manufacturer samples were prepared using an Isomet diamond wafer blade cutting process as smaller samples for mechanical testing in 3-point bend over a 10 mm span length with average thicknesses of 0.94 mm for Paradigm, 0.82 mm for the alumina and 0.36 mm for tungsten carbide.

2.2.2 Mechanical Testing

Fully articulated four-point bend fixtures were machined by MTS for A sized advanced ceramics with a 40 mm span length using ¼ point 20 mm spaced loading noses. An MTS inspection machine (858 MiniBionix) with a crosshead speed of 0.5 mm/minute mechanically tested flexural properties. Sample size estimates were performed previously so that 4 specimens from each group were tested.

$$\text{Flexural Strength}({\mathrm{\sigma }}_{\mathrm{F}}):\mathrm{\sigma }=3\text{FL}/4{\text{bd}}^2$$ [8]

$$\text{Flexural Modulus}({\mathrm{E}}_{\mathrm{F}}):\mathrm{E}=0.17{\mathrm{L}}^3\mathrm{M}/{\text{bd}}^3$$ [9]

$$\text{Flexural Strain or maximum strain in outer fibers at midspan}(\mathrm{r}):\mathrm{r}=4.36\text{Dd}/{\mathrm{L}}^2$$ [10]

ASTM C 1161-94, ASTM D 6272-00 Equation 8, ASTM D 6272-00 Equation 9 and ASTM D 6272-00 Equation 10. Specifying F-maximum load, L-span length, b-sample width, d-sample depth, M-slope of the tangent to the initial straight line on steepest part of the load-deflection curve, D-beam deflection. 3-pt bend for the small manufacturer samples: σ=3FL/2bd² and E=0.25L³M/bd³ and r = 6Dd/L².

Yield Strength was calculated from the flexural strength formula where the initial steep portion of the force-deflection curves started to deviate from linearity toward increased deflection.

Toughness analysis [5, 6] using energy methods (kJ/m²): by computer spreadsheet technology, toughness energy values can be readily calculated by summing successive complementary load deflection data points using numerical integration with the trapezoidal rule, Equation 11, where:

$$\text{Area}=\sum 1/2(\text{sum for parallel side lengths base}+\text{top})(\text{height})=\sum 1/2({\text{force}}_1+{\text{force}}_2)(\text{deflection distance})$$ [11]

With fully articulated self-adjusting fixtures, deflections may reverse so that a negative trapezoidal area is balanced by a correspondingly larger positive area.

Resilience energy was integrated by numerical methods from the load-deflection curve in the elastic region up to yield point where the initial steep straight-line slope deviates toward increased deflection for cross-sectional sample area toughness.

Work of Fracture (WOF) energy was integrated by numerical methods for cross sectional area toughness using the area under the load-deflection curve out to a maximum of 5% deviation past the peak load.

S_Ic was integrated by numerical methods for area under the load deflection curve from peak critical load to a maximum of 5% deflection past the utmost force as a function of the material cross sectional area. With small S_Ic deflections less than 1% of the total for example when no fibers are added, judgement must be used to look for deflection points that become lower due to rapid release of the platen load on the sample, such that a single load-deflection data set negatively impacting the total area may need to be discarded before summing all trapezoidal areas. Because atomic bonds at the crack tip may suddenly release potential kinetic energy through a relatively large fracture zone or cleft at critical load under the sample on the tensile flexural surface or internally under the surface, an acceleration of pressure release may hold the sample in place or even lift the sample up. Thus, zero or negative deflection data may possibly then be rearranged and considered on an individual basis instead to be added back into the final summation of trapezoidal areas if applicable to provide a positive summation value.

Flexural Critical Stress Intensity Factors (K_Ic): Interaction between flexural strength and WOF, which includes uncontrolled crack propagation, should be a good measure of processes occurring during damaging material fracture failure. The initial starter crack is considered to develop at maximum flexural strength so that the square root of the product between WOF as energy/area or N*m/area and strength as N/area provides units of MPa*m^1/2, Equation 12. Similarly, modulus and S_Ic are found as a means to evaluate K_1c using a geometrical correction factor Y that has been approximated to be 1.1215 for a fundamental edge crack as the boundary condition for initial error from sample preparation toward failure during flexural testing, Equation 13 [1, 2]:

$${\mathrm{K}}_{\text{Ic}\left(\text{sw}\right)}=1.1215{(\text{Strength}\ast \text{WOF})}^{1/2}$$ [12]

$${\mathrm{K}}_{\text{Ic}\left(\text{ms}\right)}=1.1215{(\text{Modulus}\ast {\mathrm{S}}_{\text{Ic}})}^{1/2}$$ [13]

Combining K_Ics by Equation 14 provide a value that reflects not only the starter crack condition but also crack propagation past peak load. Further, flexural K_Ic(ms+sw) testing better reflects pure Euler bending conditions with less shearing forces for accurate test data reporting.

$${\mathrm{K}}_{\text{Ic}(\text{ms}+\text{sw})}={(\text{Strength}\ast \text{WOF})}^{1/2}+{(\text{Modulus}\ast {\mathrm{S}}_{\text{Ic}})}^{1/2}$$ [14]

A second fixture with 40mm length support spans with a deepened well between support spans was fabricated so that large deflections experienced with unreinforced polymers were compared.

2.2.3 Crack-Tip Plastic Zone Defect Region

An approximation for the crack-tip plastic zone defect region (r_p) can also be determined from load-deflection data using yield strength (σ_ys) and K_Ic by Equation 15 [1, 10, 16] as:

$${\mathrm{r}}_{\mathrm{p}}=(1/2\mathrm{\pi }){({\mathrm{K}}_{\text{Ic}})}^2/{({\mathrm{\sigma }}_{\text{ys}})}^2$$ [15]

2.3 Fracture Analysis and Characterization

Nikon Microscopy provided general fracture imaging and fracture depth measurements at 20-30× magnification. Vertical fracture depths measured from the tensile failure side were normalized as a function of the sample thickness for evaluation with mechanical properties. Both lateral fractures sides were measured and averaged. X-rays: General Electric Machine at 10mamp, 65kVp and 0.4 seconds using Kodak 7.7×5.7 mm film by automatic processor. Lateral view x-rays were taken of all samples from each fiber group for general analysis particularly with regard to diffuse crack propagation beyond the fracture surfaces that tends to extend toward the neutral plane during flexural testing [5]. Scanning Electron Microscope (SEM): Representative samples were also viewed from the lateral surface to characterize the extent of material fracture depth-wise for each fiber-length group and volume-percent group. Before SEMs, samples were gold/palladium sputter coated [5, 6, 15].

3. Results

3.1 Mechanical Test Results

From the accurate derived bulk flexural test results presented in Tables 1 and 2, K_Ic could be calculated by Equation 14 and further shown below. Volume percent results were fiber length at a constant 3.0 mm and for fiber length results at a constant 28.2 V_f.

Table 2. Fiber Length Mechanical Test Results-28.2 Vf (StDev).

Fiber Length (mm)	Flexural Strength (MPa)	Modulus (GPa)	Strain (10-2)	Yield Strength (MPa)	Resilience (kJ/m2)	WOF (kJ/m2)	SIc (kJ/m2)	KIc (MPa*m1/2)	a (um)
0.0 (Z100)	117.6 (5.5)	19.5 (1.3)	.79 (.18)	95.4 (14.6)	3.03 (.99)	4.48 (.42)	0.036 (.024)	1.71 (.30)	68.8 (23.5)
0.5	113.8 (22.7)	24.0 (1.4)	.62 (.09)	92.8 (9.48)	2.35 (.57)	3.91 (1.46)	0.075 (.05)	2.17 (.71)	118.3 (46.3)
1.0	173.6 (26.0)	26.2 (0.8)	.84 (.11)	126.1 (13.0)	3.84 (.70)	8.70 (3.13)	0.097 (.054)	3.12 (.83)	101.9 (26.7)
2.0	373.9 (29.9)	34.0 (2.9)	1.34 (.21)	329.8 (47.8)	19.71 (4.21)	28.19 (3.17)	1.882 (1.005)	12.39 (3.16)	343.6 (126.4)
3.0	374.9 (11.9)	31.5 (1.2)	1.31 (.04)	343.5 (37.1)	23.31 (5.35)	30.11 (1.51)	2.400 (.534)	13.51 (1.15)	415.6 (67.1)
6.0	332.7 (55.3)	31.2 (1.2)	1.25 (.14)	252.8 (47.0)	13.04 (5.02)	25.35 (4.96)	2.047 (.347)	12.21 (.93)	431.0 (50.7)

Mechanical properties above 28.2 fiber vol % and with 6.0 mm fibers are variably less than expected by recent micromechanics [5, 6, 15] and considered to be a result of poor fiber wetting at larger fiber volume fractions and due to fiber bending with 6.0 mm fibers.

3.2 K_Ic as a Predictor for Failure

K_Ic regression with V_f is calculated from sample group averages without the correction factor from Equation 13 as K_Icms, Figure 2A, and with correction factor from Equation 14 as K_Ic(ms+sw), Figure 2B. Without the correction factor K_Ic(ms) was able to explain about 77% of the variability for V_f. With the correction factor K_Icms+sw was able to explain about 80% of the variability for V_f. By comparison resilience, work of fracture and strain energy release from the group mean Fracture Toughness Micromechanics data previously presented [5, 6] were able to explain only about 61%, 66% and, 61% respectively of the variability for V_f. Correlation Matrix Analysis of the several mechanical test variables for fracture depth was also calculated from individual sample test results rather than group mean averages by software from Statistica, Table 3. Of all the mechanical test variables, K_Ic with the correction factor was the superior predictor for vertical fracture failure explaining approximately 89% of the variability for crack depth. Thus, K_Ic with correction factor has utility as a predictor for fracture toughness failure. Still, other mechanical test variables performed well for fracture predictors in the V_f regression related study.

Figure 2.

A. Group mean K_Ic without correction factor for fiber volume percent B. Group mean K_Ic with correction factor for fiber volume percent.

Table 3. V_f Variables as Predictors for Failure by Crack Depth.

VARIABLE	r(X,Y)	r2	p
FIBER VOLUME%	-0.939338	0.882356	1.36E-13
FLEXURAL STRENGTH	-0.894001	0.799237	1.48E-10
MODULUS	-0.909038	0.826351	2.21E-11
YIELD STRENGTH	-0.862528	0.743955	3.61E-09
RESILIENCE	-0.775508	0.601413	1.25E-06
WOF	-0.809913	0.655959	1.78E-07
SIc	-0.797215	0.635552	3.81E-07
KIc	-0.88582	0.784676	3.71E-10
KIc CORRECTION	-0.943103	0.889444	6.03E-14

X-rays, Figure 3A-F, show crack depth propagation from the lower tensile flexural surface. Large cracks propagated through the materials with fibers less than 2.0 mm yet did not pass through the neutral surface with fibers 2.0 mm and longer corresponding to increases in mechanical properties even at low fiber volume fractions, see Tables 1 and 2. While reductions in strength for 6.0 mm fiber lengths are considered to be a result of fibers that are sufficiently long to bend back and not maintain linear orientation, K_Ic regression could also be calculated from individual sample data through the micromechanics for fibers using lengths from 0.0 mm to 3.0 mm at a constant 28.2V_f and compared with and without the correction factor, Figure 4.

Figure 3.

Fibers 0.0-6.0 mm, 28.2V_f, sample depth 2.0 mm: (A) 0.0 mm (B) 0.5 mm (C) 1.0 mm (D) 2.0 mm (E) 3.0 mm (F) 6.0 mm. With permission: Polymer Composites Volume 28: 311-324. 2007.

Figure 4.

(A) K_Ic without correction factor for fiber length (B) K_Ic with correction factor for fiber length.

K_I(cms) without the correction factor has slightly less variability explained at about 81% compared to K_{Ic(ms +sw)} with the correction factor at about 84% both calculated by Excel. Regarding fracture toughness prediction, K_Ic with correction factor calculated by Statistica from accurate data values using numerical integration compares well at about 83% variability explained by vertical crack depth with strength and WOF that could respectively explain about 85% and 84% of the variability for vertical fracture, Table 4. (6.0 mm fibers removed due to excessive bending).

Table 4. Fiber Length Variables as Predictors for Failure by Crack Depth.

VARIABLE	R(X,Y)	R2	p
FIBER LENGTH	-0.79292	0.628719	6.42E-06
FLEXURAL STRENGTH	-0.92373	0.853279	3.27E-10
MODULUS	-0.83302	0.693927	8.09E-07
YIELD STRENGTH	-0.87084	0.75836	6.5E-08
RESILIENCE	-0.81873	0.670314	1.79E-06
WOF	-0.91673	0.84039	7.98E-10
SIC	-0.86142	0.742039	1.3E-07
STRAIN	-0.81742	0.66817	1.92E-06
KIC	-0.89723	0.805021	6.66E-09
KIC CORRECTION	-0.90991	0.827933	1.77E-09

Predictors for fracture failure by crack depth calculated for K_Ic with correction factors appear good exceeding all other variables in the fiber volume percent analysis and compare well with strength and WOF in the fiber length analysis. Overall, standard K_Ic theory combines two mechanical test variables with Modulus and S_Ic and with the correction factor added then includes modulus, S_Ic, WOF and strength so that overall a good predictor is available for fracture toughness that incorporates four main mechanical test variables into one single variable.

3.2 Non-Reinforced Comparison Materials

K_Ic Results for isotropically-type inclined materials are presented in Table 5.

Table 5. Mechanical Properties for Isotropic-type Materials Design (StDev).

Fiber Volume Percent	Flexural Strength (MPa)	Modulus (GPa)	Strain (10-2)	Yield Strength (MPa)	Resilience (kJ/m2)	WOF (kJ/m2)	SIc (kJ/m2)	KIc (MPa*m1/2)	a (um)
Z100 3M	117.6 (5.5)	19.5 (1.3)	.79 (.18)	95.4 (14.6)	3.03 (.99)	4.48 (.42)	0.036 (.024)	1.71 (.30)	68.8 (23.5)
Alert microfiber	90.4 (19.4)	17.6 (4.9)	.69 (.05)	62.3 (12.2)	1.52 (.54)	3.23 (.70)	0.034 (.032)	1.33 (.38)	83.6 (57.3)
Polymer VinylEster	81.2 (27.6)	3.5 (0.3)	2.5 (.63)	53.2 (7.8)	9.53 (6.53)	16.19 (10.47)	0.262 (.119)	2.20 (.54)	331.7 (234.8)
Bone Cement	45.1 (11.0)	1.5 (0.3)	3.6 (0.4)	30.1 (8.3)	1.45 (0.54)	7.10 (2.14)	0.428 (.201)	1.50 (.40)	366.2 (106.3)
Paradigm 3M Corp	156.4 (16.8)	12.2 (2.0)	1.1 (.12)	129.7 (14.3)	1.44 (0.69)	1.44 (0.69)	0.055 (0.034)	1.40 (0.40)	25.6 (9.1)
Alumina Coors	231.8 (71.7)	79.6 (31.2)	0.68 (0.71)	231.8 (71.7)	0.68 (0.64)	0.68 (0.64)	0.066 (0.051)	2.59 (1.20)	37.9 (15.2)
WC 22% Co	2278.8 (40.8)	253.9 (16.2)	1.0 (0.1)	1583.1 (45.9)	5.84 (0.18)	13.46 (0.56)	0.279 (0.005)	15.68 (0.14)	15.1 (0.3)

3.4 K_Ic Literature Test Results for Comparison

Flexural bend testing for K_Ic values calculated from modulus and strain energy release and then combined with the correction factor using maximum strength and WOF could then be compared to short notch beam shear K_Ic values found in literature [17-24, 30-33]. K_Ic could also be compared with compression torsion K_Ic testing for some acrylic bone cements [25-28] and by crack fractography estimates for particulate-filled polymer Paradigm [29] Table 6. K_Ic with the correction factor obtained from reliable accurate load-deflection mechanical test data compared well overall with literature results.

Table 6. Critical Stress Intensity Factor Bulk Material Values (MPa*m^1/2).

Material	KIc(sw)a	KIc(ms)b	KIc(sw) + KIc(ms)	Literature
Resin 50:50 BisGMA:TEGDMA	1.20 (.31)	1.04 (.27)	2.24 (.48)
Resin 60:40 BisGMA:TEGDMA [17]				1.64 (.46)
Resin 97.5:2.5 BisGMA:TEGDMA	0.90 (.25)	0.53 (.16)	1.43 (.41)
Particulate-filled composite 3M Z100 [18-20]	0.82 (.04)	0.89 (.32)	1.71 (.30)	1.4-1.70
Particulate/microfiber composite Alert [20]	0.61 (.13)	0.72 (.45)	1.33 (.38)	1.25-1.57
Acrylic J&J Bone Cement [21-24]	0.63 (.16)	0.87 (.26)	1.50 (.40)	1.15-1.45
KIc compression torsion bone cement testing in literature [25-28]				1.37-1.78
Paradigm CAD/CAM Blank 3M [29-“a” length estimate]	0.53 (.16)	0.87 (.25)	1.40 (.40)	1.1 (1.3-3M)
Alumina AL 300 Coors Ceramic [30-32]	0.43 (.28)	2.16 (.94)	2.59 (1.20)	2.70-3.0
Tungsten Carbide 22% Co Basic Carbide [33]	6.23 (.07)	9.45 (.21)	15.68 (.14)	11.4-13.5

^asw-maximum strength with work of fracture;

^bms-modulus with strain energy release.

3.5 Theoretical Starter Crack (a)

Brief analysis of the theoretical starter crack “a” term by Equation 7, can be defined through the critical flaw size that develops from the stress/strain internal material potential energy primarily as WOF to initiate complete failure. The obvious results from Tables 1 and 2 show that fibers of 2.0 mm length or sufficiently above critical length (L_c) [5, 6, 15] greatly increase fracture toughness values for resilience, WOF, S_Ic and K_Ic along with the starter crack a length. L_c is at about 0.5 mm for a vinyl ester silica fiber-reinforced resin composite where all of the fiber debonds before fracture and mechanical properties are minimized for fibers until about 2.0 mm or 4 L_c [5, 6, 15]. Increases for S_Ic and K_Ic and a were clearly seen for 3.0 mm fibers at 6L_c even down to lowest fiber fractions at 5.4 V_f. Figure 3 provides excellent easy visual x-ray examples of cracks that deflect laterally below the neutral plane for composites reinforced with fibers at least 2.0 mm. In addition, fibers provide sufficient strengths with lengths of at least 2.0 mm to prevent fracture through completely with even more energy consumption by fiber pull-out and fiber bridging, Figure 5 with 6.0 mm fibers. The amount of energy measured during critical crack propagation from maximum loading by S_Ic is extremely small compared with the total amount of energy required during WOF to produce failure. But, S_Ic is only the surface energy of both crack surfaces. The vast majority of potential energy stored during applied force/deflection up to critical load is rapidly channeled out and released as high kinetic energy at velocity of about mach 2 or 600m/sec in steel [10, 34]. Energy released through the broken atomic bonds to relieve large fracture-level stress/strain on the sample propelled crack propagation damage. [34, 35,] As a result, most brittle non fiber-reinforced samples fracture to complete failure at crack initiation [35] due high velocity kinetic energy released that extends crack propagation past critical length [34]. Consequently, large differences in K_Ic results occur even within same protocols but with different substitutes for starter crack as notches or cracks that are either sharp or blunted [36].

Figure 5.

SEMs 28.2V_f with 6 mm fibers: Left 30× low magnification critical crack propagation with uniform fiber bridging starts on the lower tensile surface prevents complete fracture through the sample to provide an example consistent with theory critical crack a length at about 400 um, Table 2. Right 200× higher magnification of critical fracture for failure is held slightly by fiber bridging at about 50 um wide and far below the crack tip that is measured at approximately 2.0-4.0 um wide similar to another estimate of approximately 1 um for Ti₃SiC₂ ceramic [35].

3.6 Crack-Tip Plastic Zone Defect Region

The r_p is the region surrounding the crack tip front that has been deformed permanently beyond the material yield point. Theoretical calculations for r_p, Equation 15 with yield strength and K_Ic, applying the micromechanics for volume percent fiber-reinforced composite analysis from average values with 3.0 mm quartz fibers at about 6L_C increasing from 0V_f to 54V_f, Table 1, gave an average r_p value of 124±52 um. The K_Ic correction factor using WOF and maximum strength values that are both associated with positions above the yield point on the load deflection curve were not involved. Average values for r_p particulate-filled composite without any fibers at 0.0 V_f, however, were just 17±11 um and significantly smaller than the average for all other fiber-reinforced groups between 5.4 V_f to 54.0 V_f with an r_p at 140 ± 31 um, p < .001.

Thus, fibers might appear to be involved in large energy adsorption to minimize damage within the r_p that corresponds with fiber-reinforced fracture by long variable starter crack lengths, Tables 1 and 2, after higher critical loadings. Although fiber-reinforced fracture is manifested by a larger value for r_p, fiber crack deflections as a blunting defect minimizes the possible high velocity kinetic energy release damage beyond the crack tip in the disrupted plastic zone region further partially influenced by the polymer matrix energy consumption with a midrange r_p value of 70±42 um. Conversely, the particulate filled composites fracture through completely as released kinetic energy cannot be held into the r_p but rather continues by crack propagation to completion in smaller samples only 2.0 mm in depth, see x-rays in Figure 3.

Calculations for r_p regression could also be performed from the micromechanics by fiber length at a constant 28.2 V_f, Table 2. Analysis by coefficient of determination or R² for average r_p values of each fiber length group from 0.0 mm to 6.0 mm showed that 94% of the variability for fiber length could be explained by r_p, Figure 5. However, there was a large increase in r_p when comparing fiber lengths increasing from 1.0 mm to 2.0 mm. Between 0.0 mm and 1.0 mm length fibers, r_p values were fairly uniform at 29±13 um with an R² of just .351. With fiber lengths equal to or greater than 2.0 mm the r_p correlated remarkable well with fiber length generating an R² of .9999. In fact, all nonreinforced materials to include Alert® composite with microfibers of about 20 micrometers, 3M Corporation zirconia silicate particulate-filled composite and photocure polymer along with composites at 0.5 mm and 1.0 mm fiber lengths for low crack deflection had an average r_p of just 39±20 um.

Fiber-reinforced composites with 2.0 mm, 3.0 mm and 6.0 mm length fibers at 28.2 V_f had an average r_p of 155±50 um. Crack deflection is apparent with 0.5 mm fibers but not substantial since this is approximately one L_C where all of the fiber debonds during failure. Crack deflection would similarly be relatively low even at 1.0 mm or 2 L_C. However, as fiber lengths increase to 2.0 mm or 4 L_C and above, crack deflection should increase to help in the strain-release energy adsorption process. Actually, x-rays showed that at fiber lengths of 1.0 mm and below, crack propagation continued through the neutral surface, Figure 3, and even through the entire sample with polymer and particulate-filled or microfiber-filled composite groups. Conversely, fibers with lengths of at least 2.0 mm and greater provided sufficient crack deflection to maintain failure generally below the neutral surface with concurrent large increases in mechanical properties, Tables 1 and 2. Therefore, even still with large r_p values from much higher loading forces, fiber crack deflections would seem to limit the r_p plastic crack tip zone as larger amounts of strain energy are consumed during crack propagation across a broader expanse of material. By energy adsorption with fiber deflections, crack-type disruptions could be minimized into a larger blunted defect region. However, unreinforced polymer samples could not contain the potential kinetic energy release at any appreciable level so that fracture continued with crack propagation successively without stop or deflection through the entire material to complete failure.

4. Conclusions

Griffith-fracture theory establishes the boundary load-deflection conditions for K_Ic with modulus and strain energy release now as S_Ic to be a predictor for material fracture. Separate conditions inside the established load-deflection boundaries of modulus and S_Ic exist by using maximum strength and WOF. A new method using numerical integration is presented providing absolute bulk material mechanical K_Ic values that can either be substituted by modulus and S_Ic as an accurately derived strain energy release into the Griffith equations (for K_Ic(ms)) or within a modified equation using strength and WOF (for K_Ic(sw)) to account for correction factors associated with the starter crack where stored potential energy is lost rapidly by crack propagation in kinetic release. From highly accurate mechanical test results, both K_Ic values together (as K_Ic(sw) + K_Ic(ms)) agree well with identical isotropic material literature values. Most literature results were obtained from tests derived with excessive single edge notch beam shear test correction factors (2-3×) to account for substantial artificially created starter cracks well beyond the size for a normal edge crack that even extend into the neutral plane area where all tensile strengths are lost. General problems in K_Ic testing include very difficult crack length measurements particularly with ceramics because of relatively high modulus and low toughness properties that propagate cracks rapidly to complete fracture through the material. NAS requests for reliable fracture toughness testing is satisfied most completely by numerical integration for energy methods on resilience, WOF, S_Ic and further now with K_Ic.

Figure 6.

Group means for plastic tip zone defect region for different fiber lengths. Between 0.0 mm and 1.0 mm R² =.351. Between 2.0 mm and 6.0 mm R² = .9999.