Stress Analysis : Stress Analysis Saurabh S. Chandra
Dept. Of Conservative Dentistry
Contents : Contents Introduction to Biomechanics
Aims & Objectives of Stress analysis
Various types of Stress
Stress & Strain Curve
Tooth –restoration interface
Stress analysis techniques
Various experiments using FEA
Conclusion
References
Introduction : Introduction Traditionally Dental materials were subjected to mechanical testing only
Biological analysis was limited to soft & hard tissues
Nowadays, both Biologic & Mechanical testing of dental materials is mandatory
Slide 4: “Biomechanics is the study of the physical behavior of biologic structures and the interactions between biologic and restorative systems”
It is the application of mechanics to biologic systems
Dental Biomechanics : Dental Biomechanics Mechanical stresses (Physiologic and Pathological) that the orofacial complex are subjected to are determined
Response of the various oral tissues and restorative materials to mechanical stresses
Stress Analysis : Stress Analysis “Stress analysis is an engineering discipline that determines the stress in materials and structures subjected to static or dynamic forces or loads”
Aims & Objectives : Aims & Objectives To determine whether the structure can safely withstand the specified forces
Achieved when the determined stress from the applied force(s) is less than the ultimate tensile strength, ultimate compressive strength or fatigue strength the material is known to be able to withstand
Slide 8: Deformed Body Body What is Stress ??? When a Force acts on a body to produce deformation,
A resistance develops to this external force,
which is called STRESS
Slide 9: It is the Force per Unit Area acting on millions of atoms or molecules in a given plane of material
Stress = Force / Area
Expressed in Units of Load / Area (Pounds/in2 = PSI or N/mm2 = MPa) (Ref: Dental Materials 11th Ed. – Anusavice)
Slide 10: Engineering stress or nominal stress is defined as the ratio of the applied load to the original cross section of the specimen
Engineering stress = P (applied load) / A (cross sectional area)
True stress or actual stress is defined as the ratio of the applied load to the actual cross sectional area of the specimen
True stress = P (applied load) / A (actual cross sectional area)
Slide 11: 3 Types of Forces in Oral Cavity Continuous
(Constantly present) Impact
(One time force) Cyclic
(Multiple Impacts) The direction of the force determines the type of Stress
Types of Stress : Types of Stress 1.) Based on forces acting on the specimen:
Simple stress:
Tensile stress
Compressive stress
Shear stress
Complex stress:
Flexural stress
2.) Based on temperature changes on the specimen:
Thermal stress
Slide 13: Force
(Two Sets of Forces) Acting in the
Same straight line Acting Parallel to
Each Other Twisting or Sliding Shear Stress Directed away
From each other Directed towards
Each other Leads to Elongation Compression or
Shortening Tensile Stress Compressive Stress When a load is applied, the structure undergoes deformation
as its bonds are compressed, stretched or sheared
Compressive Stress : Compressive Stress If a body is placed under a load that tends to compress or shorten it, the internal resistance to such a load is called a Compressive Stress and it is associated with a Compressive strain
Calculated by dividing the applied force by the cross sectional area perpendicular to the force direction
Tensile Stress : Tensile Stress A Tensile Stress is caused by a load that tends to stretch or elongate a body and it is always accompanied by Tensile strain
Tensile stress is generated when structures are flexed
Shear Stress : Shear Stress It is produced by a twisting or torsional action on a material. A shear stress tends to resist the sliding on a portion of a body over another
Shear stress is calculated by dividing the force by the area parallel to the force of direction
Slide 18: Mechanical properties of a material describe its response to Loading
Most clinical situations involve complicated 3-D loading situations; it is common to describe the loads in terms of a single direction (Compression, Shear, Tensile)
Combinations of these can produce Torsion (twisting) or Flexion (Transverse bending)
Slide 19: During loading, bonds are not compressed as easily as when they are stretched
Materials resist compression readily and are stronger in compression than in tension
As loading continues, the structure is ultimately deformed
Strain : Strain Strain is defined as change in length per unit initial length
Strain (ε) is deformation (▲L) per unit of length (L)
Expressed as inch/inch or cm./cm.
The Stress-Strain Curve : The Stress-Strain Curve With a constant increase in loading, the structure is ultimately deformed
At first, the deformation (Strain) is reversible – Elastic Strain
With increased loading, there is some irreversible strain which results in permanent deformation – Plastic Strain
Slide 22: Elastic Limit Strain Stress Ultimate Strength Linear straight line till Elastic Limit
Afterwards, its concaves towards the strain axis
Ultimately leads to a fracture
Slide 23: Elastic Strain – the deformation that is recovered upon removal of an externally applied force or pressure
Plastic Strain – the deformation that is not recoverable when an externally applied force is removed
Slide 25: The point of onset of plastic strain is called the Elastic Limit (Proportional Limit, Yield Point)
It is indicated on the stress – strain curve as the point at which the straight line starts to become curved
Continuing the plastic strain leads to Fracture
The highest stress before fracture is the Ultimate Strength
Slide 26: Elastic limit of a material is defined as the greatest stress to which a material can be subjected to, such that it returns to its original dimensions when force is released
Materials that undergo extensive plastic deformation before fracture are called Ductile (in Tension) and Malleable (in compression)
Materials that undergo very little plastic deformation are called Brittle
Elastic Modulus (E) : Elastic Modulus (E) Also called Young’s Modulus or Modulus of Elasticity
It describes the relative stiffness or rigidity of a material, which is measured by the slope of the elastic region of the stress – strain graph
It represents the amount of strain produced in response to each amount of stress
Eg. Ceramics have a higher ‘E’ than polymers, which means ceramics are stiffer
Slide 29: Elastic modulus has a constant value that does not change and it describes the relative stiffness of a material
The Elastic modulus of Enamel is higher than that of Dentin
Depending on the area of the tooth, studies have suggested that the value may be 3 to 7 times
Slide 30: Stiff Flexible Shape of S-S curve and magnitude of stress & Strain helps in classifying materials
For eg. If the longitudnal portion of the curve is closer to the long axis, material is stiff Strain Stress
Stress – Strain plot for Enamel & Dentin : Stress – Strain plot for Enamel & Dentin
Slide 32: Enamel is stiffer and more brittle than dentin
Dentin is more flexible and tougher and is capable of sustaining significant plastic deformation under compressive loading before it fractures
Clinical Significance : Clinical Significance When a load is applied to a tooth, it is transmitted through the material giving rise to stresses and strains. If these exceed the maximum value the material can withstand, a Fracture results
Clinical Application : Clinical Application The most useful properties of a restorative material are Modulus of elasticity (E) and Elastic limit
A restorative material should be very stiff so that under load, its elastic deformation is minimal
An exception to this is in Class V cavities, where Microfill Composites are used – They should be less stiff to accommodate for tooth flexure
Slide 35: When selecting a restorative material, the clinician must bear in mind the stress level during function
This should not exceed the Elastic Limit
If the stress level is beyond the elastic limit, a resulting deformation is likely to occur which may cause failure at some point of time
Slide 36: The elastic limit can be determined by various “Hardness Tests”
Various hardness tests include:
Mohs
Rockwell
Knoop
Brinell
Vickers
Tooth –Restoration Interface : Tooth –Restoration Interface The biomechanical behavior of a restored tooth is of immense significance to the clinician
A standard biomechanical unit involves -
Tooth structure
Restorative material
Interfacial zone (Interface)
Slide 38: In a normal tooth – loads are transmitted to dentin through enamel
Dentin undergoes a small amount of deformation (Tooth Flexure)
This amount of strain is proportional to the amount of load applied on the tooth
Slide 39: A restored tooth transmits stress differently
Enamel is not continuous and its resistance lowered
Therefore, restorations should be designed in a manner in which the stress is distributed to the dentin rather than enamel
Once in dentin, they act in a similar way as a normal tooth
Slide 41: Stress transfer occurs between
Restorative material to tooth structure
Tooth structure to PDL
From teeth to underlying bone
Most common analysis is the stress transfer at the Tooth – Restoration Interface
Slide 42: Stress Analysis…
Stress Analysis of Dental Restorations : Stress Analysis of Dental Restorations Mechanical properties of a material used in dental restoration must be able to withstand the stresses and strains caused by the repetitive forces of mastication
It is necessary to use designs that do not result in stresses or strains that exceed the strength properties of a material under clinical conditions
Historical perspective : Historical perspective The main method for determining stress values and distributions has been by means of indirect experimental techniques
Hoppenstand and McConnell used a model simulation to study the mechanical failure of Class I type amalgam restorations
Mahler et al used a similar technique to investigate design aspects of Class II restorations
Traditional methods of experimental stress analysis,
include transmission and reflection, 2-D & 3-D photoelasticity, brittle lacquers and electrical resistance strain gauge techniques
Slide 45: Techniques used for Stress analysis are:
Theoretical: Use mathematical formulation and solution of the resultant equations
Experimental: Involves measurements of various types made directly on the structures of interest or through the use of modeling of structure
Slide 46: Theoritical techniques:
Finite Element Analysis (FEA)
Experimental techniques:
Strain gauge
Photoelasticity
Fine Element Analysis (FEA) : Fine Element Analysis (FEA) Developed in 1943 by Richard Courant
Used the “Ritz method” of numerical analysis and variational calculus to obtain approximate solutions to vibration systems
FEA is a computer stimulation technique used in engineering analysis. It uses a numerical technique called Finite element method
A common use of FEA is for the determination of stresses and displacements in mechanical objects and systems
Slide 48: Finite element method, is a modern technique of numerical stress analysis
Has the advantage of being applicable to solids of irregular geometry and heterogeneous material properties
It is therefore ideally suited to the examination of the structural behavior of teeth
Slide 49: Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle
Computer-aided engineering (CAE) is the application of computer software in engineering to evaluate components and assemblies in stress analysis (FEA / FEM)
Slide 50: 2-D, FEM solution for a magnetostatic configuration (lines denote the direction of calculated stress and colour - its magnitude) 2D mesh for the image above (mesh is denser around the object of interest)
Slide 51: This technique simulates actual continuous structures with discrete element mathematical representations
The basic concept of this technique is the visualization of the actual structure, as an assemblage of a finite number of structural elements connected at a finite number of points
The finite elements are formed figuratively cutting the original structure into segments. For two-dimensional applications, triangles of various sizes and shapes are usually the finite elements of choice
Slide 52: Each element retains the mechanical characteristics of the original structure
A numbering system is required to identify the elements and their connecting points, called nodes and a coordinate system must be established to identify uniquely the location of the nodal points
A large number of simultaneous linear equations are computer generated, which establish compatibility within each element
Slide 53: Two types of analysis that are used:
2-D modeling
3-D modeling
2-D modeling conserves simplicity and allows the analysis to be run on a normal computer; Results are less accurate
3-D modeling, however produces more accurate results but can run only on specialized computers
How Does Finite Element Analysis Work? : How Does Finite Element Analysis Work? FEA uses a complex system of points called nodes, which make a grid called a mesh
The mesh is programmed to contain the material and structural properties, which define how the structure will react to certain loading conditions
Nodes are assigned at a certain density throughout the material depending on the anticipated stress levels of a particular area
Slide 56: Regions that will receive large amounts of stress usually have a higher node density than those, which experience little or no stress
Points of interest may consist of: fracture point of previously tested material, corners, complex detail, and high stress areas
The mesh acts like a spider web, whereby in each node, there extends a mesh element to each of the adjacent nodes. This web of vectors is what carries the material properties to the object, creating many elements
Steps in Finite Element analysis : Steps in Finite Element analysis There are 3 phases in FEA:
Pre-processing – defining the finite element model and environmental factors to be applied to it
Analysis solver – solution of finite element model
Post-processing of results using visualization tools
Slide 58: Pre-processing
This is the first step in FEA, whereby construction of a finite element model of the structure to be analyzed is made
This can be in either 1D, 2D, or 3D form, modeled by line, shape, or surface representation (nowadays 3D models are predominantly used)
The primary objective of the model is to realistically replicate the important parameters and features of the real model
Slide 59: The simplest mechanism to achieve modeling similarity in structural analysis is to utilize pre-existing digital blueprints, design files, CAD models, and/or data by importing that into an FEA environment
After the finite element geometric model has been created, a meshing procedure is used to define and break up the model into small elements
In general, a finite element model is defined by a mesh network, which is made up of the geometric arrangement of elements and nodes
Slide 60: Nodes represent points at which features such as displacements are calculated
FEA packages use node numbers to serve as an identification tool in viewing solutions in structures such as deflections
Elements are bounded by sets of nodes, and define localized mass and stiffness properties of the model
Elements are also defined by mesh numbers, which allow references to be made to corresponding deflections or stresses at specific model locations
Slide 61: Analysis (computation of solution)
The next stage of the FEA process is analysis
The FEM conducts a series of computational procedures involving applied forces, and the properties of the elements which produce a model solution
Such a structural analysis allows the determination of effects such as deformations, strains, and stresses which are caused by applied structural loads such as force, pressure and gravity
Slide 62: Post-processing (visualization)
These results can then be studied using visualization tools within the FEA environment to view and to fully identify implications of the analysis
Numerical and graphical tools allow the precise location of data such as stresses and deflections to be identified
Slide 63: Visualisation
Results can be studied using visualisation tools within FEA based computers
To view and identify the implications of the analysis, Numerical and Graphical tools are used
These tools allow the precise location of data such as Stresses and deflections
Slide 64: Results of FEA
FEA predicts failure of materials which results due to unknown stresses by showing their problem areas
This method of product design and testing is far superior to the manufacturing costs that would accrue if each sample was actually built and tested
Slide 65: Difficulties in using FEA in dentistry
The difficulty associated with the models elaboration, for they present different shapes depending on the tooth to be analyzed
The difficulty involved in obtaining the mechanical properties of the tooth’s constituent materials: enamel, dentin, cementum, pulp etc.
Furthermore, little is known about the contact areas between such materials and their degree of influence on the mechanical behavior of the tooth as a whole
Experimental Techniques : Experimental Techniques Various experimental techniques can be used to predict stresses or strains and measure the mechanical response of a structure to simulated or actual loads
Use either models of the structure of interest or the actual structure, depending on the technique employed
Strain Gauge
Photoelasticity
Strain Gauge : Strain Gauge A strain gauge is a device used to measure deformation (strain) of an object
Invented by Edward E. Simmons and Arthur C. Ruge in 1938
The most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern
The gauge is attached to the object by a suitable adhesive
Slide 68: A strain gauge is a long length of conductor arranged in a zigzag pattern on a membrane
When it is stretched, its resistance increases. Strain gauges are mounted in the same direction as the strain and often in fours to form a full 'Wheatstone Bridge'
Slide 69: They use the principle that when a certain electrical resistance is subjected to an object, it produces strain
Tension produces an increase in resistance; compression causes a decrease in resistance
Therefore, if such a strain gauge were bonded to the surface of a structure under a load, monitoring the resistance changes would yield knowledge of the strain characteristics at that point
An experimental analysis of stresses using Strain gauge : An experimental analysis of stresses using Strain gauge In vitro study evaluated the stress transfer of different post and core systems to the cervical part of the artificially created flared root canals, by using strain gauge
The post–core systems investigated were: (a) cast post–core system without resin reinforcement, (b) cast post–core system with resin reinforcement, (c) pre-fabricated
post and resin core with resin reinforcement
Slide 71: The post–core systems which were cemented on simulated roots were subjected to a load applied at an angle of 45° to the long axis of the simulated roots
The strain gauges which were cemented to the cervical part of simulated roots were connected to the data acquisition module to measure and record the changes in strain data
Slide 74: Specimens restored with resin reinforcement either with cast post–core or pre-fabricated post and resin core transferred the stress to the cervical part of the artificial roots at a rate lower than conventional cast post–core system
It was concluded that the resin reinforcement of root canals before post–core applications reduces the stresses at the cervical part of the root surfaces
Photoelastic stress analysis : Photoelastic stress analysis Based on the property of some transparent materials to exhibit colorful patterns when viewed with polarized light
These patterns occur as the result of alteration of the polarized light by internal stresses into two waves that travel at different velocities
The pattern that develops is consequently related to the distribution of the internal stresses and is called Photoelastic effect
Slide 76: To use this special characteristic, a model of the structure of interest must be fabricated in the right dimensions and proportions
The model should be made from a transparent material capable of exhibiting a photoelastic response
The stresses that develop in the model as the result of the applied loads can then be visualized by examining the model with polarizing filters
Automated Photoelastic Stress Analysis : Automated Photoelastic Stress Analysis Automated photoelasticity uses a computer to calculate principal strain differences and directions without the need to count fringes or rely on the subjective interpretation of fringe colours. Furthermore, photoelastic coatings allow for full field strain measurements to be made on a structure under load.
Slide 80: The general procedure for photoelastic analysis involves bonding a special plastic coating onto the structure, shining polarized light onto the plastic, and then analyzing the resultant images.
Advantages : Advantages Stresses can be determined in models of complicated three-dimensional shapes, such as the oral structures thereby facilitating the location and magnitude of stress concentration
Stresses resulting from complex loading conditions, such as forces of mastication and forces produced by restorative appliances can also be determined
Slide 82: Photoelastic stress analysis has developed into a powerful, accurate and widely used technique in engineering and industry to analyze areas of excessive stress concentrations
It has facilitated the design of complicated structures and machinery and has had wide application in dentistry
Two-dimensional Photoelastic Stress Analysis of Traumatized Incisor : Two-dimensional Photoelastic Stress Analysis of Traumatized Incisor In this study, stress of traumatized incisor and the effect of stress on tooth and alveolar bone was studied with two-dimensional photoelasticity
Two homogeneous two-dimensional maxillary central incisor models were prepared
Loads were applied to the labial side of incisal edge and middle third of the crown at angles of 45° and 90°
It was observed that stress was increased on teeth and alveoler bone when load was applied 90° on labial side of incisal edge
Various Experiments using Finite Element Analysis : Various Experiments using Finite Element Analysis
Finite Element Stress Analysis of a Normal Tooth : Finite Element Stress Analysis of a Normal Tooth Enamel presents a similar behavior to ceramics, being a fragile material and crystalline (hydroxyapatite crystals)
Anisotropic characteristic of enamel
Researches try to present differences of enamel’s mechanical behavior when submitted to occlusal loads depending on the load directions in relation to the enamel (Spears 1993, Las Casas 2003)
Slide 87: Motta et al analyzed the influence of enamel anisotropy on stress distribution in a sound tooth (Neves, 2003)
In order to describe the results, paths were created
Results showed that there is a difference in stress distribution between the isotropic and anisotropic enamel models
FEA on a Mandibular Premolar : FEA on a Mandibular Premolar Normal mastication generates considerable reactionary stresses in teeth and their supporting tissues
Enamel is assumed to be isotropic, has greater stiffness over that of the dentin
The masticatiory forces tend to “flow” around the enamel cap although the dentin core remains lightly stressed
This is seen to be the cause for isotropic or orthotropic enamel under single or two-point loading.
Slide 90: As the modular ratio between the enamel and dentin decreases, the distribution of force between two components becomes less unequal
Enamel near the DEJ is highly stressed because the reacted forces have to flow into and through this thin wedge of tissue for them to be transmitted into the root of the tooth
Slide 91: Therefore evident that restorations inserted into the cervical region of the teeth can be subjected to direct contact stresses of mastication
This may be the reason for pain often experienced by patients who have received cervically placed restorations
High stresses are generated in the fissure under masticatory type loading
Although these stresses are not as large in magnitude as those induced near the DEJ, they have consequences
Slide 92: Tensile stresses tend to pull the enamel prisms apart in this region and may thereby assist the attack of caries in the fissures of premolar and molar teeth once the chemical demineralization of the enamel has been initiated
The distraction of the cups would also tend to open up the margins of any restoration placed between the two occlusal contact points
An increase in the marginal crevice would obviously encourage the separation of cariogenic material into these spaces and consequently assist in the breakdown of cavity margins placed in this area
First molar with amalgam restoration : First molar with amalgam restoration The model is divided into a number of triangles
The smaller triangles are located in areas of great interest
The ability of various types of cement bases to support the amalgam was also studied
Slide 94: The stress induced in amalgam restoration was from 4-5 times higher when the amalgam was supported by 2mm zinc oxide eugenol cement base compared with an equal thickness of zinc phosphate cement base.
Zinc oxide eugenol cement base does not function as a rigid material and induces larger displacement and even thin layers induced significant changes in the stress induced in amalgam.
The fracture of the amalgam is influenced more by the modulus of elasticity of the base material then by the compressive strength of base
The cement base must have moduluis of elasticity equal to that of restorative material.
Teeth with direct composite restorations : Teeth with direct composite restorations “In vitro” analysis was done in teeth with Class I cavity preparation by using different axial wall inclinations (convergent or parallel), and by inserting composite resin
Teeth were sliced in the middle and the defects on the tooth-restoration interface were measured. Based on the data, a 2D model was designed for each cavity
The objective of this study was to evaluate the influence of axial wall inclinations in relation to defect presence and stress distribution when submitted to occlusal loading
Slide 96: Results show different stress distributions in
the two analyzed models
Stress analysis of an upper central incisor restored with different posts : Stress analysis of an upper central incisor restored with different posts The effect of different anatomic shapes and materials of posts in the stress distribution on an endodontically treated incisor was evaluated
Compared three post shapes (tapered, cylindrical and two-stage cylindrical) made of three different materials (stainless steel, titanium and carbon fibre on Bisphenol A-Glycidyl Methacrylate (Bis-GMA) matrix)
Two-dimensional stress analysis was performed using the FEA. A static load of 100N was applied at 45° inclination with respect to the incisor’s edge
Slide 99: The stress concentrations did not significantly affect the region adjacent to the alveolar bone crest at the palatine portion of the tooth
Stress concentrations on the post ⁄ dentin interface on the palatine side of the tooth root presented significant variations for different post shapes and materials
Post shapes had relatively small impact on the stress concentrations while post materials introduced higher variations on them
Stainless steel posts presented the highest level of stress concentration, followed by titanium and carbon ⁄Bis-GMA posts
3-D analysis : 3-D analysis 3D models of natural and abfraction lesions were constructed and submitted to FEM analyses (ABAQUS) under physiological and on-lingual-cusp-only loading conditions. The choice for lingual cusp loading was made based on 2D model results
Conclusion : Conclusion Stress analysis techniques are invaluable for manufacturers of dental materials as they help in evaluating critical stress levels of various materials
They help in evaluating the mechanical properties of dental materials under laboratory conditions and also give a three dimensional view
Their accuracy has been a point of confrontation for many years
References : References Sturdevant’s Art & Science of Operative Dentistry
Phillips Dental Material – 11th Ed.
Applied Dental Materials – Mc’ Cabe (8th Ed.)
Dental Materials – Richanrd van Noort (2nd Ed.)
Materials of Restorative Dentistry – Parameswaran
Journal of Oral Rehabilitation
Journal of Operative Dentistry
Journal of Prosthetic Dentistry
Journal of Restorative Dentistry