2.1.

Three-Dimensional Digital Photoelastic Analysis: Stress Distribution Pattern

Three-dimensional digital photoelasticity was conducted in four stages. In the first stage, models were prepared using a photoelastic liquid resin. In the second stage, the models were subjected to postcure cycle. In the third stage, the models were subjected to a stress-freezing procedure. In the fourth stage, the stress frozen models were sectioned along different planes of interest and analyzed in a circular polariscope using a four-step phase shift technique.

The models were prepared using a liquid resin system supplied as two liquids consisting of a resin and a hardener (PLM-1, Measurements Group, Inc.). During model preparation a human mandible with anterior dentition obtained from an adult cadaver was sectioned distal to the lateral incisors on the left and right side. The central incisors were then carefully extracted from the socket and replicating molds of the mandible and the central incisor were prepared using silicone rubber base impression material (PROVIL, Bayer Dental). Subsequently, 12 parts by weight of hardener was taken with 100 parts of resin and mixed well until the temperature (exothermic reaction) rose to 52 °C. At that temperature, the liquid epoxy was poured into the previously prepared mold.

The initial polymerization of resin was completed in 24 h after pouring into the mold. The models were next subjected to a postcure cycle. The postcure cycle consisted of heating the models along with the silicone rubber mold in an oven for up to 80 °C at a rate of 5 °C/h. The plastic was then maintained at this temperature for 2 h and then cooled to room temperature at the rate of 5 °C/h. Following this cycle, the photoelastic models were ready for stress freezing.¹⁵ The entire manipulation of the photoelastic material (PLM-1) was carried out in accordance to the manufacturer’s instruction. These steps ensured proper curing of photoelastic plastic and aid in achieving the prescribed properties of PLM-1.¹⁶

Models of mandibular central incisor and supporting bone models were individually prepared for this study (Fig. 1), and this facilitated simulation of periodontal ligament at the tooth-bone interface. Models were prepared based on the relative modulus of elasticity of different dento-osseous structures. Photoelastic liquid plastic (PLM-1) with elastic modulus of 2.9 GPa was used to prepare the tooth and the supporting bone models. A layer of chemically cured silicon rubber, 0.35 mm thick, and an elastic modulus of 0.2 MPa, was used to simulate the periodontal ligament. A polyester reinforced composite sheet 0.15 mm thick simulated the root cementum. This sheet had an elastic modulus that was similar to that of photoelastic models but with higher resilience. The earlier thickness for silicon rubber and polyester composite sheet was chosen to replicate the original anatomical dimensions of periodontal ligament and cementum. Since the objective of this study was to evaluate the stress distribution pattern in structural dentine, the model materials were chosen keeping in mind the relative elastic modulus difference between the anatomic structures.¹⁷

Figure 1

Three-dimensional photoelastic models of (a) the tooth and supporting bone separated, (b) the tooth-supporting bone models interfaced by silicon rubber to simulate periodontal ligament, (c) the cross sections at the cervical and apical regions of the tooth, and (d) the schematic diagram showing different tooth sections and anatomical planes.

Three teeth and supporting bone models, and a calibration disk were prepared from a single batch of photoelastic liquid plastic. The models and the calibration disk were next subjected to a stress-freezing procedure. The stress-freezing procedure was an important step in three-dimensional photoelasticity. During stress-freezing, the models were subjected to loads at a specific elevated temperature. The load induced stress patterns were then frozen within the model during the cooling phase. In this study the model was gradually heated to 125 °C, and was loaded. A loading jig was designed and fabricated to hold the anatomically scaled models within an oven. An experimental load of 10 N was applied along the long axis of the tooth model at 125 °C. This load was chosen due to the lower stiffness of the model material at higher temperature. Further, a pilot study showed that this load was sufficient to produce optimum number of fringes for the sectional model analysis. The temperature of 125 °C was maintained for 2 h to facilitate uniform softening of the entire model material. This stage was followed by a cooling phase. The entire stress-freezing procedure was carried out at a rate of 5 °C/h. The value of the stress-optic coefficient at the stress freezing temperature was determined from the calibration disc (14 kPa/fringe m).

After the stress-freezing cycle, the tooth model was separated from the bone model and was cross sectioned along different regions of interest at the cervical third and apical third of the tooth [Fig. 1(d)]. Sectioning of the models was carried out immediately after stress freezing using a diamond-impregnated wheel, mounted on a milling machine with distilled water as a coolant. The sectioned, stress-frozen models were then placed between the first and the second quarter-wave plate of a circular polariscope, and the fringe patterns were recorded using a charge coupled device (CCD) camera (Fig. 2). In digital photoelasticity, rotating the analyser at 0°, 45°, 90°, and 135° with respect to the polarizer induced the four phase steps [Figs. 3(a), 3(b), 3(c), and 3(d)]. These four phase-shifted images were evaluated using a phase unwrapping algorithm, in order to obtain the wrapped phase map [Fig. 3(e)]. Further details regarding the digital photoelastic fringe analyzis is presented elsewhere.¹⁷

Figure 2

Schematic diagram of a digital photoelastic experimental arrangement.

Figure 3

Shows the phase shifted images of a calibration disc obtained by rotating the analyzer’s axis of polarization with respect to the polarizer’s axis of polarization for (a) 0°, (b) 45°, (c) 90°, and (d) 135°, in a circular polariscope (e) the phase wrapped image obtained from the four phase shifted images and (f) the phase wrapped map of the tooth and the supporting bone (sagittal section).

The stress distribution obtained from the wrapped phase map look similar to a conventional fringe pattern, but displayed phase information rather than intensity information of the fringes. Phase unwrapping was carried out along selected lines in the wrapped image to make the fringe modulation continuous and to obtain details on the absolute nature and magnitude of stress distribution. The phase unwrapping method is based on modulation ordering and relies on the intensity modulation in pixels.¹⁷ In this study the stress patterns along the faciolingual plane (sagittal plane) were evaluated before sectioning the models, whereas the stress patterns along the cross sections were evaluated after sectioning the models.

2.2.

Digital Moire´ Interferometric Analysis: Strain Gradients

Specimens for the moire´ interferometric analysis were prepared from three permanent noncarious mandibular incisor teeth, stored in phosphate buffered saline solution immediately following extraction. The specimens were transilluminated prior to testing to exclude possibilities of cracks and were tested within 3 weeks of extraction. The teeth specimens for the moire´ investigation were prepared by grinding the mesial and distal surfaces on wet emery papers of grit size 180, 400, 800, and 1000 to obtain parallel-sided, sagittal sections with a uniform thickness of 2.5 mm.

Moire´ methods utilize grating as a deformation-sensing element. To prepare the specimen grating, a high frequency grating from a mold was transferred onto the surface of the specimen using a thin layer of epoxy adhesive (PC-10, Measurements Group, Raleigh, NC). This specimen grating was interrogated using a virtual reference grating formed by the interference of two mutually coherent beams incident on the specimen plane at a fixed angle (Fig. 4). Moire´ fringes result from the interference between the deformed specimen grating and the virtual reference grating. These fringes represent contour maps of in-plane displacement fields and were analyzed using image processing software. Further details on the specimen grating preparation are presented in Post etal.¹⁸

Figure 4

Schematic diagram of the digital moire´ interferometry experimental arrangement.

The experiment consisted of the following four stages.

A high frequency cross grating (f=1200 lines/mm) was replicated on one faciolingual surface (sagittal side) of the tooth section using epoxy adhesive at room temperature. The replication was done in such a way that the grating lines were parallel and perpendicular to the long axis of the tooth. The distance between the adjacent grating lines is referred to as the pitch of the grating, which in this case was 0.833 μm. The specimen with the replicated grating was mounted on a horizontal loading jig that permitted compressive loading of the tooth specimen. The loading jig is designed with a load cell, which was connected to a digital display that aided in monitoring the applied load. The loading jig was subsequently positioned in the moire´ interferometer (Fig. 4).

The moire´ interferometer was used to visualize the resulting load induced deformation fringes. The moire´ interferometer consisted of two mutually coherent light beams from a diode laser (λ=670 nm) which were incident on the specimen grating at an oblique angle and generated a virtual reference grating of 2400 lines/mm [corresponding to a pitch (P) of 0.417 μm]. This virtual reference grating interacted with the deformed specimen grating to produce the moire´ fringe patterns. A high resolution CCD camera (Electrim 1000HR, NJ) was used to digitize and record the fringe patterns obtained for further analysis.

Moire´ fringes represent contours of displacement components in the direction perpendicular to the lines of the grating. Hence, during our experiments the surface displacements along the x axis, known as U field and that along the y axis, known as V field were obtained by rotating the specimen in the interferometer to enable the specimen grating lines perpendicular to the x and y axis to interact with the virtual reference grating individually (Fig. 5). During experiments compressive loads of 10, 20, and 30 N were applied at the incisal edge, along the long axis of the tooth, and the moire´ fringes were acquired.

Figure 5

Typical image showing (a) the U field (deformation in the axial direction) and (b) the V field (deformation in the lateral direction) moire´ fringe patterns.

Strain is a geometric quantity, which depends on the relative movements of two or three points in a body. Strains in experimental mechanics are technically classified as: normal and shear strains. A normal strain is defined as the change in length of a line segment between two points divided by the original length of the line segment. A shear strain is defined as the angular change between two line segments, which are originally perpendicular. The normal and the shear strain gradients for each load, along selected lines of interest in the cervical and apical dentine (Fig. 5) were determined as follows.

From the U field fringe pattern the normal strain (ε_x) in x direction (axial direction/along the long axis of the tooth) is given as:

Similarly ΔU/Δy can be written as

From the V -field fringe pattern, the normal strain (ε_y) in the y direction can be deduced as

The shear strain (γ_xy) is then given as

3.1.

Three-Dimensional Digital Photoelastic Analysis: Stress Distribution Pattern

It was found that the applied compressive loads resulted in a distinct bending stress at the coronal and cervical region of the tooth model. The bending stress resulted in low stress at the inner region, which increased towards the facial and lingual surfaces [Fig. 6(a)]. Bending stress is produced in structures subjected to eccentric load (loads acting away from the line of symmetry). Bending results in a transition of stress (or strain) from tensile to compression, passing through a zero at neutral axis. A conspicuous reduction in bending was observed at the apical region of the tooth [Fig. 6(b)].

Figure 6

Typical graphs showing the stress distribution pattern in the sagittal plane at the (a) cervical and (b) apical dentine.

Figures 7(a) and 7(b) are phase wrapped images showing the stress distribution at the cross sections from the cervical and middle third of the tooth model. The cervical cross section displayed conspicuous distribution of bending stresses along the faciolingual plane [Figs. 7(c) and 7(d)]. These stresses increased from the inner region towards the outer surfaces. The facial surface displayed higher stress compared to the lingual side. The cross section at the apical dentine showed less signs of bending and revealed more uniform distribution of stresses. The magnitude of stresses at the apical region was lower than that in the cervical region.

Figure 7

Shows (a) the typical phase wrapped map at the cervical region (cross section) of the tooth model and (b) the typical phase wrapped map at the apical region (cross section) of the tooth model. (c) The typical graph showing the stress distribution pattern at the cervical region (cross section) of the tooth model and (d) the typical graph showing the stress distribution pattern at the apical region (cross section) of the tooth model.

3.2.

Digital Moire´ Interferometric Analysis: Strain Gradients

The U field (axial strain) and V field (lateral strain) moire´ analyses showed that the compressive loads resulted in a distinct distribution of normal and shear strains along the axial and lateral directions in the dentine [Figs. 5(a) and 5(b)]. The outer aspect of the coronal and cervical dentine exhibited prominent axial strains during compression. The U field analysis showed that the initial loads on the tooth produced conspicuous distribution of axial strains at the cervical dentine. These strains at the cervical dentine were both normal and shear in nature [Figs. 8(a) and 8(b)]. The facial aspect of the cervical dentine displayed shear strains adjacent to the cemento-enamel junction [Fig. 8(b)]. The root dentine, which was apical to the cervical region, did not display noticeable strains for loads less than 20 N. However, at 30 N load there was a uniform distribution of normal strains in root dentine [Fig. 8(c)]. In root dentine, there was no shear strain in this direction. The normal axial strains in the apical dentine were lower than the normal axial strains in the cervical dentine [Fig. 8(d)].

Figure 8

Typical graphs obtained from the U field analysis showing (a) the normal strain and (b) the shear strain distribution in the axial direction at the cervical dentine, (c) the normal strain distribution in the axial direction at the apical region, and (d) the normal strains in the axial direction at the cervical and apical dentine.

The V field analysis showed strain distribution in the lateral direction (perpendicular to the long axis) in the dentine. The cervical dentine displayed predominant distribution of lateral strains. The lateral strains in the cervical dentine were both normal and shear in nature [Figs. 9(a) and 9(b)]. The root dentine demonstrated uniform distribution of the normal strains in this direction [Fig. 9(c)]. There were no shear strains in the lateral direction in root dentine. The normal lateral strains in the apical dentine were lower than the normal lateral strains in the cervical dentine [Fig. 9(d)].

Figure 9

Typical graphs obtained from the V field analysis showing (a) the normal strain and (b) the shear strain distribution in the lateral direction at the cervical dentine, (c) the normal strain distribution in the lateral direction at the apical region, and (d) the normal strain distribution in the lateral direction at the cervical and apical dentine.