Fracture resistance of Ceramic-Polymer hybrid materials using microscopic finite element analysis and experimental validation

Abstract

The objective of this paper is to elucidate the response to contact stresses of Polymer Infiltrated Ceramic Network (PICN) using the microscopic viscoplastic finite elements, validated by clinically relevant in vitro tests. A feldspathic ceramic material, namely Vita Mark II, is an interconnected structure infiltrated with the polymer (PMMA). Axisymmetric finite element microstructure models are reconstructed from two-dimensional images of a PICN microstructure. Viscoplastic finite element analysis (FEA) with various degrees of microscopic damages occurring over contact is performed. The force-displacement responses obtained from FEA are validated with Hertzian contact tests. Finite element results for force-displacement, stresses and strains in each phase are discussed. We hypothesize that the resistance to fracture of PICN can be further improved by microstructural tailoring. The experimental evidence suggests that a composite material is both more resistant to displacement under load and more resistant to crack initiation and propagation, as hypothesized. Further parametric study on the effects of various volume fractions of two phases in PICN is done to provide some insight on increased contact damage resistance of PICN as well as potential optimization of microstructures.

1. Introduction

The dental industry, as is true of most modern industries, is in a never-ending search to produce a product that better serves the customer. The introduction of hybrid polymer ceramic materials for use in restorative and prosthetic dentistry is the dental industry’s latest attempt at just that (Zhang and Kelly, 2017). For years, polymer and ceramic based materials were used in restorative dentistry with each material having its own advantages and disadvantages (Ferracane, 2011). The discovery of the interconnection of the two material networks in polymer infiltrated ceramic network (PICN) materials showed encouraging signs toward combining the advantages and limiting the disadvantages of the two materials in experimentation (Coldea (2014), Coldea et al. (2013, 2014), He and Swain (2011), Giordano (1995, 2000)). Physical testing has been done on various PICN materials to evaluate the material’s ability to resist contact stress and prevent crack initiation and crack propagation (Coldea et al. (2013, 2014), El Zhawi et al. (2016)). However, it is believed that finite element analysis could be used as a predictor of material behavior to further support results from physical testing, while providing increased depth and flexibility of testing method and subsequent results, and thus, a more detailed evaluation of material behavior (Tribst et al. (2020), Ausiello (2019)). Understanding damage resistance of various test specimens will allow the industry to better understand the effects of microstructural interconnection and more easily establish the most desirable combination of the two material networks.

The method of finite element analysis, as highlighted in this study, will provide the dental industry with an approach to a complementary material response analysis to the traditional method of laboratory testing (Vaidya and Kim, 2013). This study focuses on the use of composite materials, specifically PICN materials, for restorative dental applications. Recent studies from our laboratories and elsewhere have demonstrated that PICN material (Enamic, Vita Zahnfabrik) exhibits some promising properties in resistance to contact and flexural damage (Coldea (2014), Coldea et al. (2013, 2014), He and Swain (2011), Facenda et al. (2018), Ongun et al. (2018), El Zhawi et al. (2016)). In vitro testing and corresponding calculation of the mechanical properties of the tested PICN material was reported in Della Bona et al. (2014). However, the physics behind physical phenomena in PICN has not been fully understood.

This study highlights the importance and the advantages of the use of finite element analysis (FEA) to optimize material microstructure. We hypothesize that the resistance to fracture of PICN can be further improved by microstructural tailoring, which can be assisted by microscopic viscoplastic FEA. Both physical test results and computer simulation methods were used to analyze the effectiveness of various combinations of dental materials. Physical testing methods such as the Hertzian indentation test, the single edge V-notched beam test and the three point bending test are performed to understand the response of various porcelain ceramic, polymer, and PICN materials for contact stress and damage response under representative loading. Viscoplastic finite element analysis is then employed to further examine the material responses. The three material compositions were examined for performance in a modeled Hertzian indentation situation to provide the investigator and the reader with a deep understanding of various materials’ microstructural responses. A comparison of such responses will then commence to highlight the advantages and disadvantages of various materials and how it aligns to current research. Use of finite element analysis will subsequently be examined for applicability and accuracy for use in simulation of the response of composite restorative dental materials.

2. Materials and experiments

Twenty plate specimens each for Enamic, Mark II, and PMMA (12 × 14 × 5 mm) were prepared (see Figure 1). PICN (Enamic) consists of a 75 vol% of feldspathic ceramic (Vita Mark II) network, which is infiltrated with a low-viscosity acrylate polymer (PMMA). All plate specimens were polished on both top and bottom surfaces using an automatic grinding/polishing machine (EcoMet 30, Buehler, Lake Bluff, IL, USA). Polishing were carried out using a sequence of diamond abrasive discs 15 μm, 10 μm, 6 μm, 3 μm, and 1 μm, with water irrigation. Contact damage resistance of PICN, feldspar, and PMMA were evaluated using the Hertzian indentation method. Rigid tungsten carbide (WC) balls of diameter D = 3.16 mm were used as an indenter, simulating an average radius of contact cusps in posterior teeth (Krejci et al. (1999), Kim et al. (2008), Kim et al. (2008)). The elastic modulus of tungsten carbide balls is E = 640 GPa (Ren and Zhang (2014)). Critical loads for the onset of cone cracks were recorded for PICN and feldspar, while that for significant plastic deformation was determined for PMMA.

Figure 1.

Instron 5566 universal testing machine (left) and prepared samples (right).

Table 1 provides basic material properties used in the microscopic FEA such as Young’s modulus and the Poisson’s ratio. As known, feldspar has much higher elastic modulus than PMMA. These properties are used in continuum finite elements for each material.

Table 1.

Elastic material properties used.

Material	Young’s modulus	Poisson’s ratio
Feldspar (Vita Mark II)	54.50 GPa	0.23
Polymer (PMMA)	3.31 GPa	0.32

Hertizian contact tests have been done for PICN, feldspar and PMMA. The measured maximum load capacities for the first two systems are given in Table 2 and are used as critical forces from which cracks and damage start to initiate.

Table 2.

Hertizian contact test results.

Test Specimens	Load capacity using a WC indenter (D = 3.16 mm)
PICN (75% feldspar + 25% PMMA)	303.3 (N)
Feldspar	104.4 (N)

3. Image-Based finite element microstructure models

Axisymmetric finite element microstructure models were reconstructed from two-dimensional (2-D) images of a PICN microstructure. The 2-D images were obtained using a scanning electron microscope (Figure 2). These images are of the real PICN microstructure having 75 vol% feldspar and 25 vol% PMMA compositions.

Figure 2.

Microstructure of PICN. (a) A scanning electron microscope image, and (b) a cropped (168 × 240 pixels) image of (a).

The PICN image contains a subset of 168 (width) and 240 (height) pixels with each pixel of 0.485 μm size. A set of 8 × 8 pixels are assembled into one finite element to increase computational efficiency. An element set has 30 × 21 elements that are generated based off of the representative microstructural image. The real specimen is a rectangular plate with dimensions of 12 mm × 14 mm × 5 mm of which one quarter is modeled due to symmetry (see Figure 3).

Figure 3.

(a) Schematic of axisymmetric specimen of 7 mm × 5 mm subject to Hertzian indentation of the radius = 1.58 mm; (b) finite element mesh. The subsurface zone (size of 1.34 mm × 1.91 mm) underneath the indenter is discretized with real microstructure of PICN.

The model is defined in the axisymmetric coordinates that consists of repeated 16 subsets in the radial(r) and axial (z) coordinates, respectively. A MATLAB program was developed to generate elements with such microstructural details as an input file for ABAQUS. A viscoelastic finite element analysis was carried out using the commercial FE software ABAQUS (Abaqus, 2020).

4. Viscoelastic and viscoplastic properties of PMMA

Figure 4 shows a one-dimensional schematic of this material model, with the elastic-plastic network at top and the elastic-viscous network at bottom (Abaqus, 2020). A viscoelastic material has elastic and viscous components as shown. The viscosity of a viscoelastic material has a strain rate dependence on time. When a viscoelastic material like PMMA is subject to stress, parts of the long polymer chain change positions. This rearrangement is called creep. In addition, PMMA shows a viscoplastic response when subjected to compression. Thus the material behavior for PMMA comprises elastic, plastic, and viscous components.

Figure 4.

Two-way viscoplastic formulations used.

The ratio of the elastic modulus of the elastic-viscous network K_V to the total instantaneous modulus K = K_P + K_V is denoted by f = K_V/K. The viscous behavior is assumed to be governed by the Norton-Hoff rate law given by (Abaqus, 2020)

$${\dot{\epsilon }}_V^v=A{\sigma }_V^n$$

The subscript V denotes field quantities in the elastic-viscous network. This form of the strain rate law may be used by choosing a time-hardening power law for the viscous behavior and setting a material parameter m = 0 that relates the deviatoric stress and the viscopalstic strain rate (Fig. 4). The total strain is given by

$$\epsilon ={\epsilon }^{el}+\left(1-f\right){\epsilon }_P^{pl}+f{\epsilon }_V^v$$

where ε^el is the elastic strain

$${\epsilon }^{el}=f{\epsilon }_V^{el}+\left(1-f\right){\epsilon }_P^{el}$$

ε^pl is the plastic strain and ε^v is the viscous strain. Six material parameters need to be calibrated: K_p and K_v; the initial yield stress σ_y; the hardening coefficient H; and the Norton-Hoff rate parameters, A and n. A static uniaxial tension or compression test determines the long-term modulus, K_p; the initial yield stress; and the hardening coefficient. The instantaneous elastic modulus, K, can be measured from the initial elastic response of the material. Note that the long-term steady-state behavior of the elastic-viscous network under a constant strain rate applied, ${\dot{\epsilon }}_o$, is a constant stress given by

$${\sigma }_V=A^{-\frac{1}{n}}{\dot{\epsilon }}_o^{1/n}$$

It can be assumed that the hardening modulus is negligible compared to the elastic modulus. Then the steady-state response of the overall material is given by:

$$\sigma =A^{-\frac{1}{n}}{\dot{\epsilon }}_o^{1/n}+{\sigma }_y+H\epsilon$$

where σ is the total stress for a given total strain ε. Viscoplastic stress-strain relations are obtained from Mulliken and Boyce (2006). Viscoplastic true stress–true strain relations used in this analysis include a constant yield strength of 95 MPa for the plastic strain between 0 to 0.5 and a linearly increasing strength from 95 MPa to 145 MPa between 0.5 and 1.0. The relevant parameters are A = 0.0002, n = 4, m = 0 and f = 0.50005 (Abdel-Wahab et al., 2017).

5. Finite element analysis of hertzian contact

A FE model is developed to incorporate microstructural details of two phases. The microstructure of PICN is imported to the subsurface zone (1.34 × 1.91 mm) of the FE model. First, an FE analysis is conducted for a baseline model with no damages embedded for feldspar, PMMA, and PICN. Then the models for feldspar and PICN are modified to simulate the effects of various levels of damages such that inter-element cracks are introduced between ceramic-ceramic elements and between ceramic and polymer elements. Three steps of validation analyses are performed in order. First, feldspar and PMMA specimens are validated. Then, PICN is validated by calibrating the level of local damages in the subsurface zone.

5.1. Fe validation for feldspar and PMMA

Figure 5 shows validation of FE analysis for the indentation load vs. displacement for feldspar and PMMA. The maximum load capacity for the ceramic before damage progression was 104 N from the test. Cone crack starts to form in feldspar after this damage initiation load. FE solutions were first verified with theoretical solutions for ceramic material with no damage assumed. As these FE results are compared with test data, the force-displacement response is overestimated because of no damage assumption. Due to complexity of modeling damage phenomena underneath the contact zone, a certain percentage of inter-element (element-to-element interface damage) damage is assumed to be distributed over the subsurface zone. Various degree of damage in the subsurface ceramic zone is assumed to change from 10% to 31% with the indentation load magnitude such that FE predictions agree with test data for feldspar. The percentage denotes the number of damaged element edges to all ceramic-phase element edges. Note that one element has four edges and one damaged edge will create a crack-like damage in between two elements. Note that the validate FE analysis for feldspar shows that the contact force reaches about 1400 N at1.0 mm with accumulative damage of about 31%.

Figure 5.

Validation of FEA for indentation load vs. displacement with test data for feldspar (in comparison with dotted test data) and PMMA (green curves). The maximum load capacity for the ceramic before damage progression was 104 N from the test.

On the other hand, using two-way viscoplastic formulations, FE predictions for PMMA are in excellent agreement with test results with no damage assumed. It shows that no damage occurs in PMMA for the range of deformations tested.

Figure 6 shows a representative von Mises stress contour in feldspar predicted by FEA. Note that stress-governed damage is developed and highly concentrated under the contact zone in the red, yellow and green zones. Cone fracture is anticipated to initiate from the top surface and propagate at a 20° inclined angle normal to the maximum principal stress directions (Lawn (1998); Zhang et al. (2005)). Note that the present analysis aims to provide qualitative data for the contact resistance corresponding to a certain level of damage accumulations. When it comes to modeling PICN, it is quite challenging to perform crack growth simulations of multiple micro cracks in two different material phases.

Figure 6.

Feldspar - von Mises stress (MPa) contour.

For PMMA as shown in Figure 7(a), the magnitude of the von Mises stress is much lower than that in its ceramic counterpart due to its viscoelastic and viscoplastic properties. Much larger plastic strain is predicted as expected. The resistance of PMMA to large deformations is seen with the equivalent plastic strain of 0.6 (see Figure 7(b)) beneath the contact zone. This explains why PMMA provides deformation resistance and ductility to PICN. The FEA predictions for contact force (909 N) and vertical displacement of the indenter tip (U2 = 0.500 mm) match very well with test results, respectively, 902 N, U2 = 0.5436 mm. Not like a linear-elastic material such as feldspar, permanent deformations are observed as shown in Figure 7(c) in PMMA due to visco-plasticity (U2 = 0.359 mm) even after the indenter is removed. The vertical deformation of 0.141 mm is recovered upon unloading.

Figure 7.

PMMA – (a) the von Mises stress (MPa) contour; (b) the equivalent plastic strain (PEEQ) contour at the end of load application; (c) permanent deformations (U2) after removal of the load.

5.2. Fe validation for PICN

Figure 8 shows validations of FE analysis for the indentation load vs. displacement with test data for PICN. The maximum load capacity before damage progression was 303 N from the test. For PICN, damages are inserted in between ceramic elements (from 0% to 20% damage) assuming the feldspar-PMMA and PMMA-PMMA interfaces are stronger. Various degree of damage in the subsurface zone is assumed to change from 0% to 20% with the indentation load magnitude such that FE predictions agree with test data for PICN. Compared with test data (red squares), the initial FEA with no damage included provides the upper limit for the force-displacement response. Various levels of damages such as 5%, 10%, 12.5%, 15%, 17.5% and 20% are included in ceramic phase element-to-element boundary. As shown in Figure 8, FEA in comparison with test results shows that damage accumulates as the contact force and corresponding indentation deformation increases. For the given range of force-displacement data, FEA predicts that PICN would have damage of 20% in the ceramic phase by the time that contact force reaches about 1400 N at 0.1 mm. As such, PICN shows better damage resistance than feldspar with damage of 31% with the similar magnitudes of contact force and displacement (see Figure 5).

Figure 8.

Validation of FEA for indentation load vs. displacement with test data for PICN with progressive damage from 0% to 20%. The maximum load capacity before damage progression was 303 N from the test.

For the case of no damages included, the magnitude of the von Mises stress is higher underneath the indented surface as seen in Figures 9(a) (2 D view) and 9(b) (3 D View), which indicates typical subsurface compressive yielding and potential damage. Note that microstructures are embedded in a repeated fashion. Higher stresses are recorded in ceramic parts, while higher strains are in polymer parts. Reasons for higher contact damage resistance that are observed in PICN composites are twofold. First, as seen in figures, damages due to indentation are not concentrated in one area (such as the area under the contact zone in Figure 6) but dispersed in an inhomogeneous PICN medium. Second, the polymer possessing higher failure strains tends to prevent an unstable crack growth propagating mostly through the brittle ceramic phase, which may deflect the crack path to the interface.

Figure 9.

PICN – (a) 2 D view of von Mises stress (MPa) contour. Microstructures are embedded in a repeated fashion. (b) 3 D view. No damage was introduced.

Figures 10(a) and 10(b) show 2 D and 3 D logarithmic strain contours, respectively, for the non-damaged PICN. As seen, the strains are dispersed into the two-phase medium and the maximum strain reaches 1.0 in the PMMA phase under the contact zone, and is gradually reduced as the region gets further away from the contact zone.

Figure 10.

PICN – (a) 2 D view of logarithmic strain contour. (b) 3 D view of logarithmic strain contour. No damage was introduced.

As the load increase further, the above no-damage assumption is not reasonable. So damage was introduced step-by-step into the PICN composite. Figures 11(a) and 11(b) show the von Mises stress and the maximum principal strain contours, respectively, in the subsurface zone. We observe that the unique two-phase microstructure of PICN is seen to diffuse stresses, strains and damages into the interconnected network, thus increasing its toughness and resistance to crack initiation and propagation.

Figure 11.

PICN – (a) von Mises stress (MPa) contour; (b) Maximum principal strain contour. 20% damage was introduced in the ceramic phase.

6. Parametric study for PICN with superior fracture resistance

To perform sensitivity study of PICN with different volume fractions, the FE model with 75% feldspar and 25% PMMA was modified to change the volume fractions of ceramic and polymer. Note that the length of interface between ceramic and polymer within a subsurface damage zone in an FE model depends on the volume fraction of materials. The far-field zone is considered as a homogeneous medium with effective properties. As expected, 70%/30% specimen has largest interface lengths among the three (see Table 3).

Table 3.

Interface length of subsurface damage zone in FE models.

70/30 PICN	75/25 PICN	80/20 PICN
460 mm	410 mm	342 mm

Figure 12 shows the potential damage envelope for the force-displacement response of PICN with the progression of damage from 0% to 20%. The solid blue, red and green lines represent 80%/20%, 75%/25% and 70%/30% specimens, respectively, with no damage initiated and progressed. The dotted blue, red and green lines represent 80%/20%, 75%/25% and 70%/30% specimens, respectively, with 20% damage assumed. Note that 80%/20% and 75%/25% PICNs show better damage resistance with damage of 20% than feldspar with damage of 31% with the similar magnitudes of contact force and displacement (see Figure 5). As shown, the 80%/20% specimen has highest contact force resistance, but is expected to possess lowest ductility due to highest ceramic phases. The initial contact damage resistance of the 80%/20% model is a little higher than the 75%/25% specimen. As damage is accumulated up to 20%, its damage resistance become closer to and still a little higher than that of the 75%/25% model. On the other hand, the 70%/30% model shows much less contact resistance and more ductility due to more polymer phases. However, the present analysis cannot provide the quantitative information that cone cracks could be easier to form in one composition than in the other compositions. The comparative and qualitative numerical analysis of the three specimens supports that the 75%/25% specimen may be the most optimal combination among the three which gives both high contact damage resistance and good ductility. Further experimental study on the effect of volume fractions will need to be carried out validate this numerical observation.

Figure 12.

FE predictions for contact force-displacement response envelope for 80%/20%, 75%/25% and 70%/30% PICN.

7. Conclusions

Finite element analysis proved to be a valuable tool in predicting the overall damage resistance of various restorative dental materials. Use of viscoplastic finite element analysis allows the researcher to further support experimental evidence and helps to optimize design parameters. We hypothesized that the resistance to fracture of PICN can be further improved by microstructural tailoring. The experimental evidence suggests that the interconnection of material microstructures in polymer-infiltrated ceramic-network (PICN) materials takes advantage of the favorable properties of polymer and ceramic materials while reducing the adverse properties of the aforementioned materials. It supports the fact that a composite material is both more resistant to displacement under load and more resistant to crack initiation and propagation, as hypothesized. In particular, the PICN material with 75%/25% volume fractions showed favorable damage resistance quantified with respect to force-displacement and stress-strain responses compared to homogenous polymer and ceramic materials and to 80%/20% and 70%/30% PICNs. The numerical study presented throughout this paper has proven to be a success and will be a valuable source of information to future researchers and the dental industry as a whole.