there have been questions about its class i versus ii versus iii classification , or it being a lever in the first place     [ 1416 ] and the loading assumptions .  
but it is the lever model , coupled with the bending beam analogy of the mandible that form the foundation of the generally accepted ( longitudinal ) stress distribution pattern depicted in figure 1(a )     [ 1821 ] and its variant , figure 1(b )     [ 18 , 20 , 22 , 23 ] .
the lever is an oversimplified structural representation of the jaw , so biomechanical concepts derived from it should be suspect .   in the present context , the lever 's intrinsic vertical - only occlusal force is its primary drawback .  
thus , the principal purposes of this paper are to demonstrate , with basic analytical engineering mechanics , but barebones mathematics , the shortcomings of the lever / beam model - based concepts that relate to the surgery of the mandible .   more specifically , the central issues involve ( 1 ) the relationship between occlusal force direction and the associated stress distributions within an intact mandible and ( 2 ) the extrapolation of those stress fields to a plated fractured mandible .

 figure 2(a ) is a partial ( because dimensions irrelevant to this discussion are omitted ) free - body diagram ( fbd ) of the frame- ( versus lever- , figure 1 ) idealized mandible .  
an fbd , used in equilibrium analysis , shows all external loads ( forces and moments ) that act on an isolated object ( i.e. , the mandible or a portion of it ) of interest .   for the purposes of this project
, the crucial difference between the lever ( figure 1 ) and the frame model is that in the latter , the direction of force vector t is not necessarily vertical , figures 2 and 3 .  
the fbd in figure 2(b ) is of an imaginary or real ( i.e. , fractured ) segment of the mandible that is anterior to the arbitrarily defined ( at distance  from the occlusal contact ) circle ,  , on the approximated centroidal axis ( dashed line ) of the mandible .  
 and  represent the internal shear and normal ( perpendicular ) forces , respectively , and  represents the internal moment .
 ,  , and  symbolize the net internal loads at the interface ( real or imaginary ) that are necessary to maintain the static equilibrium of the anterior segment when occlusal force t acts on it .   applying static equilibrium conditions to the fbd in figure 2(b ) yields 
 ( 1)forcesvertical=0:ty=0  or  =ty,forceshorizontal=0:+tx=0  or  =tx,moments abouto=0:tytx=0or  =tytx . 
					
the right hand sides of the equations are known because they are specified or calculated .   for the purposes of this project
, it is important only to recognize that  and  are determined by ty and   tx , respectively , and that  is a function of both ty and tx , as well as anatomy (  ) and the horizontal distance (  ) between the section and the occlusal contact .
the analysis is based on the range of occlusal force directions represented in figure 3(a ) .  
occlusal contact can occur between a maxillary cusp distal incline and a mandibular cusp mesial incline ( t0 , t1 , and t2 ) , a maxillary cusp mesial incline against a mandibular cusp distal incline ( t4 ) or flat - plane occlusion ( t3 , also in figure 1 ) .  
( according to classic friction principles , assuming frictionless contact , the direction of a contact force between surfaces must be perpendicular to their common tangent .  
t 
 04 can be characterized by the positions of their lines - of - action ( loa ) relative to the circle , .   t1 passes above ( or equivalently , behind ) it , thus requiring a clockwise ( cw ) moment ,  , for equilibrium .  
equilibrium dictates that  must be counterclockwise ( ccw ) with t2 , t3 , and t4 , because they pass below or in front of . the second relevant characteristic of a t is the direction of its horizontal component .  
t0 , t1 , and t2 are directed posteriorly ; therefore , for equilibrium ,  must be to the right .   because t3 ( the lever analogue ) is vertical ,  = 0 .  
thus , the 5 ts generically represent the permutations of the loa orientations relative to  and the horizontal force component directions .  
it is emphasized that ( 1 ) ) and figures 24 ( essentially ,  ,  , and  ) apply identically to the imaginary anterior segment of an intact mandible and to an actual fractured - off section , but that is where the similarity ends .  
the physical manifestations of  ,  , and  in the two scenarios are fundamentally distinct .  
the system that is equivalent to  ,  , and  is produced by the stresses within the bone .   in the plated mandible
, it is produced by the loads in the plates and by contact between the fractured bone surfaces .  
elementary engineering beam theory is used to determine the stress systems ( figure 5 ) that are equivalent to a specific set of +  ,  , and  ( figure 4 ) .  
( the identical engineering principles are implicitly behind the generally accepted as  obvious  figure 1 stresses . )   a uniform compression is the equivalent to an anteriorly directed  ( c in figures 5(a ) , 5(b ) , and 5(c ) ) .  
similarly , uniform tensile stress , t , is the equivalent to the posteriorly directed  in figure 5(e ) .  
( there is no uniform tension or compression associated with t3 , figure 5(d ) , because t3 , being vertical , requires that  = 0 for equilibrium . )  
the equivalents of a cw  ( figure 5(b ) ) and a ccw  ( figures 5(c ) , 5(d ) , and 5(e ) ) are stress gradients with compression - on - top / tension - on - bottom ( c on top / t on bottom ) and the opposite , respectively .  
( a uniform stress , without a gradient , is sufficient for t0 equilibrium , figure 5(a ) . )   for completeness , the shear stress equivalent of  is included in figure 5 , but as generally done in the literature , it is henceforth ignored . as noted above , figures 24 and ( 1 ) ) also apply exactly to a plated fractured mandible ( figure 6(a ) ) .  
however , the system that is equivalent to a specific set of   +  ,  , and  ( figure 4 ) is provided by the loads ( hu ,  vu ,  mu ,  hl ,  vl ,  ml ,   , and  ) identified in the fbd of the plated fractured segment in figure 6(b ) not the distributed bone stresses at the imaginary section ( figures 1 and 5 ) .
h (  ) , v (  ) , and m (  ) are the unknown forces and moments acting within the upper and lower plates directly overlying the bone fracture , figure 6(b ) .  
 and  are representative of the normal ( perpendicular ) and shear contact forces , respectively , on the fractured bone surface if the two segments touch .  
( according to basic friction theory , the maximum  , max =  , where  is the coefficient of friction between bones .  
this is likely to influence the results of experimental studies in which the mandible analogue plastic has a different  than bone . )
similar to the derivation of ( 1 ) ) , three independent simultaneous equations of static equilibrium , based on the fbd in figure 6(b ) , can be obtained by summing forces in the vertical ( 2 ) and horizontal ( 3 ) directions and by summing moments ( ( 4 ) about the tooth contact point 
 ( 2)ty++vu+vl=0 , 
 ( 3)tx+huhl=0 , 
 ( 4)duhuvu+mudlhlvl+ml=0 . 
							 because there are fewer independent equations than unknown forces and moments (
only tx and ty are known ) , this ( statically indeterminate ) problem can not be solved using only the principles of static equilibrium .  
fortunately , solving the problem is not a goal .   equations ( 2)(4 ) serve as explicit evidence of the complexity of plating biomechanics .
as consequence of loa position ( figure 3 ) , t1 produces a counterclockwise ( ccw ) moment ( rotation ) about the circle ; t0 does not cause a rotation ; t2 , t3 , and t4 produce clockwise ( cw ) rotations .   for the equilibrium of the intact or fractured mandible ,
these moments , and t 's horizontal and vertical components , must be opposed by the internal loads  ,  , and  , respectively , figures 2(b ) and 4 and ( 1 ) ) . in the intact mandible ,  ,  , and  are expressed as equivalent stress distributions , figure 5 .  
the sums , according to the superposition principle , of those individual stress distributions ( excluding the -associated shear stress ) yield the net longitudinal tension / compression stress gradients shown in figure 7 .   (
the result in figure 7(d ) is identical to that depicted in figure 1(a ) . ) in a plated mandible , the load system that is equivalent to  ,  and  is supplied by the loads on the plates and by the contacts between bone fragments .  
but even a simplified plated assembly ( figure 6 ) is far too intricate to solve with the methods of basic static equilibrium .
equations ( 1)(4 ) clearly indicate the complex relationships between what is known / given and what is unknown .
three - dimensional ( 3d ) numerical     [ 26 , 27 ] and experimental   models often lead to questions about the prevailing lever - based ( tot / cob ) stress distribution dogma .  
however , those previous criticisms involve the location of occlusal forces ; the focus of this paper is on the direction of those forces
.   specifically , the lever - based two - dimensional ( 2d ) model , figure 1 , consists of one force component direction ( vertical ) and one moment component direction ( perpendicular to the page by the right - hand rule ) .   the model in figure 2 , the basis for the presented analysis , adds the horizontal force component .  
this is the most general 2d model that is possible , and it is used to demonstrate serious deficiencies in the lever method without having to resort to the complexities of a 3d approach . for convenience , and/or perhaps because of the lever model legacy , studies have generally been limited to occlusal forces in one ( usually vertical , t3 ) direction .  
( in some experimental setups , the occlusal plane is slightly canted , so with a vertical force , the effect is similar to t4 . )  
but there is no reason to believe that a bolus of food would necessarily elicit a vertical , or any particular , occlusal force direction .  
if , instead , crown - crown contact is assumed , then the vertical force assumption restricts analyses solely to frictionless flat - plane or edge - to - edge occlusions .  
this study is framed in the context of crown - crown contacts , so the specified contact angle ( i.e. , cusp incline angulation or incisal guidance ) generally defines a nonvertical orientation of the occlusal force .   some readers may prefer to attribute nonvertical occlusal forces to the interactions of muscle activity with the presence of a bolus of food .   in either case , there is no compelling justification for the vertical occlusal force simplification , and as demonstrated , occlusal force direction is a critical determinant of the mechanical environment ,  ,  , and  , within a mandible .  
the fbd in figure 3(b ) can serve the same purpose as the fbd in figure 2(b ) .  
however , the former is presented to emphasize that without the ( entirely superfluous ) mandible outline , this is actually a mundane static equilibrium problem in which a force , t , is being applied at a point located in a specified ( by  and  ) position relative to another point ,  , where the reactions  ,  and  are of interest .  
therefore , for this aspect of the analysis , the only things that matter are the direction of t 's loa and its position relative to ( i.e. , distance from ) .   for specific quantitative values of  ,  , and  , the magnitude of t would also be needed .  
but for the qualitative analysis , the relative magnitudes of  ,  , and  are sufficient , and because this is a linear model , these relative magnitudes do not change with changes in t magnitude .  
another purpose for presenting figure 3(b ) as an alternative is to illustrate that the human mandible drawing in figure 2(b ) could easily be replaced with a nonhuman mandible , a machine part , and so forth . without any effects on  ,  , and . the 5 occlusal forces , t04 ,
but instead of a continuum of directions , it is more practical to consider categories of force directions defined by the positions of their loa 's relative to  , combined with the directions of their horizontal components .  
for example , whenever the loa of a force passes through  , generically represented by t0 , the results presented herein for t0 apply .   naturally , the angulation of t0 ( defined by  and  ) would be different for incisal versus molar contact , nonetheless , qualitatively , the stress distributions would be identical .  
this concept is already being taken for granted with t3 , because the stress distribution in figure 1 is not specified for any particular contact location .  
t2 's loa is bounded by t3 and t0 , shaded region in figure 5(c ) , so the associated stress distribution can be seen to morph , left to right in figure 7(c ) , from the tot / cob of t3 ( figure 7(d ) ) to the complete compression of t0 , ( figure 7(a ) ) .  
similar trends , but with less defined endpoints , occur with t1 and t4 because their loa 's are bound only on one side by t0 and t3 , respectively . the stress distribution within the body of the mandible
, there is a decreasing tension gradient from the gingival height , and near the mid - level , there is a reversal to an increasing gradient of compression toward the inferior border .  
this tension on top compression on bottom ( tot / cob ) stress distribution is consistent with the inherent vertical occlusal force of the lever .  
but in reality , in general , there must also be a nonzero horizontal bite force component , figure 3 , whenever inclined planes ( cusps ) contact each other .   and
as demonstrated , its presence has the potential to profoundly alter the stress distribution within the mandible from the generally accepted tot / cob ( figures 1 , 5(d ) , and 7(d ) ) to , as an example , exactly the opposite ( figure 7(b ) , left ) .
the depictions of the relative magnitudes of the stress distributions in figures 5 and 7 are not drawn to scale because the mathematical complexities of the governing equations , necessary for quantitative results , are being circumvented and because such detail would be unnecessarily obfuscating .  
serendipitously , the simplifications are perfectly suited to the mathematics - minimized approach of this paper , because the critical nuances of figure 7 can be examined without resorting to rigorous mathematics .
if t acts through the circle , t0 , then the longitudinal stress in the body of the mandible is entirely a uniform compression , figures 5(a ) and 7(a ) .  
( as illustrated in figure 4(a ) , t0 requires that  = 0 for equilibrium , hence the uniform stress . )  
that , of course , is contrary to the tot / cob lever - based stress distribution , figures 1 and 7(d ) .
the sums of the 1 associated uniform compression ( c ) and the cw 1 associated c  t gradients of t1 , figures 4(b ) and 5(b ) , can combine to produce 3 different net stress distribution patterns , figure 7(b ) .   because both 1 and 1 are associated with compression in the top part of the mandible ( c and c , respectively ) , there is no doubt that the top part will be in compression .   in the bottom part of the mandible , 1 is associated with compression ( c ) , but 1 is associated with tension ( t ) .  
thus , depending on the relative magnitudes of c and t , the net longitudinal stress in the bottom part of the mandible can be tensile , zero or compressive ( left , middle , and right , respectively , in figure 7(b ) ) .   to determine which of the three stress patterns reflects the actual stress , the governing beam equations would have to be solved .  
but for the present purposes , it suffices simply to note that all 3 potential t1 results are at variance with the lever 's tot / cob stress distribution . 

t 
 2 ( figures 4(c ) and 5(c ) ) , like t1 , produces a uniform compression consistent with 2 , but because 2 is ccw , its associated gradient is t  c.   so , when these 2 longitudinal stress distributions are superimposed , figure 7(c ) , it is certain that the bottom part of the mandible is in compression , but depending on the position of t2 's loa within the shaded region in figure 5(c ) , the top part can be in tension , stress - free , or in compression .   more specifically , as t2 approaches vertical , its longitudinal stress distribution ( figure 7(c ) , left ) looks more - and - more like the ( vertical ) t3-associated stress ( figure 7(d ) ) .  
as t2 's direction approaches t0 , its stress distribution , figure 7(c ) right , starts to resemble the t0 produced stress , figure 7(a ) . 

t 
 3 ( the lever replica , figures 4(d ) and 5(d ) ) is vertical , so there is no 3 equivalent stress .  
its tot / cob stress distribution , figures 1 and 7(d ) , is entirely the equivalent of the ccw 3 .   and
finally , t4 ( figures 4(e ) and 5(e ) ) is different from the others , because the 4 equivalent stress is a uniform tension .  
combined with the ccw 4 's t  c gradient , the top portion of the mandible will be in tension , but the bottom portion can be in tension , stress - free , or in compression , figure 7(e ) .  
thus , of the 11 possibilities depicted in figure 7 , only 3 ( left in figure 7(c ) , figure 7(d ) , and right in figure 7(e ) ) are in concert with the lever 's tot / cob longitudinal stress distribution .  
( oral surgeons are intimately familiar with the biomechanical principles at work here .   during extraction ,
the tooth can be considered as a vertical analogy to the mandible that is loaded as in figures 5(b ) and 5(c ) .  
when the tooth is luxated back and forth ,  are generated , and to reduce the associated tensile longitudinal stress in the root , a compressive ( intrusive ) force is applied concurrently , thereby reducing or eliminating the tension , analogous to the right sides of figures 7(b ) and 7(c ) turned 90 degrees .   a lever model of extraction would not account for the critical intrusive force component (  ) that prevents tension - induced root fracture . ) for the plated mandible , figure 6(b ) , there are an insufficient number of equations to solve for all unknowns .  
this statically indeterminate problem can be numerically modeled , typically with fea     [ 2936 ] , but whatever the solution may be , it must satisfy ( 2)(4 ) .  
thus , as evidenced by the equations , everything is intertwined in a complex manner .  
furthermore , if the fracture were at an angle , or if the plates were not horizontal or parallel , then ( 2)(4 ) would be further complicated by trigonometric functions , different  values for the upper and lower plates , and so on .  
this is clearly not a problem that is amenable to commonsensical analysis , especially with intuition that is based on the nonapplicable lever paradigm of the intact mandible .
consider plates , ab and cd , secured with single screws at their ends , figure 8(a ) .  
abdc , figure 8(b ) , forms a 4-bar linkage in which b and d are constrained to move along arcs of circles that are centered at a and c , respectively .  
therefore , if for any reason b moves to b , d must move to d , because the bone dimension , bd (= bd ) , remains unchanged .  
thus , with virtually any occlusal force , these plate constraints dictate a downward displacement with a ccw rotation of the anterior segment from bd to bd , hence the inferior border distraction .  
when the concepts and terminology from the intact mandible are applied to this outcome , it is ascribed to compression on top and tension on the bottom     [ 26 , 37 ] .  
but , in fact , the top plate , ab , is in tension and cd , the bottom plate , is in compression , and except for the compression at the bone - bone contact on top , there is no tension ( which would be impossible anyway at a fracture ) or compression in bone anywhere along the fracture .
thus , contrary to popular notion , the enlarged gap at the inferior border is not caused by tension ; on the contrary , it is caused by compression in cd. there are , of course , many other 4-bar linkage configurations that would produce different interfragmentary displacements .   because plate ab , as the example , is secured by single screws at its ends , it can only transmit pure compression or tension  it is a 2-force member .  
( if , in figure 6 , the top plate was affixed with single screws at both ends , then hu  0 , but vu = mu = 0 . )  
accounting for friction ( between bone / plate or screw / plate ) , or even just one additional screw , would immensely complicate the problem by necessitating the inclusion of the shear forces ( i.e. , vu  0 in figure 6(b ) ) and the bending moments ( i.e. , mu  0 in figure 6(b ) ) in the plates .   the relative bone movements would then depend mostly on plate deformations caused by those forces and moments .  
clinically , however , it would be advantageous to conceptualize plate - imposed constraints in terms of linkages rather than the intact mandible paradigm .  
( neither is realistic , but in this context , the intact idealization is entirely irrelevant and misleading . )
although the focus is on plates , similar discussions would pertain to other modes of fixation .  
for example , in a sagittal split osteotomy , the transverse shear stresses within the bicortical screws and bone - bone friction produce the  ,  , and  equivalent system .   as with plating ,
a mandible in function , intact or plated , is a complicated 3d statically indeterminate structure that is subjected to the complex interactions of variable anatomy and highly changeable muscle and occlusal forces .   and
, for many reasons , obvious and subtle , a plated fractured mandible is an entirely different and more complex load bearing structure than an intact mandible .  
if for no reason other than the impossibility of tensile stresses acting across a break , the tot / cob stress distributions illustrated in figures 1 , 5 , and 7 are not possible at , or near , a fracture .   according to saint venant 's principle , these stress distributions can be present only at some distance away from plated areas .  
nevertheless , the figure 1 tot / cob representations are often invoked explicitly     [ 18 , 20 , 22 , 23 , 31 ] or implicitly in matters of fracture repairs .
although the demonstrated analytical method is inadequate for the solution of these problems , it is the best instructional approach , and it is conceptually far more realistic than the ubiquitous lever model .  
this analysis is appropriate for conceptualizing mandibular biomechanics and for casting doubt on the status quo . in conclusion ,
( 1 ) the lever - based tension - on - top / compression - on - bottom ( tot / cob ) stress gradient should not be assumed in the intact mandible .  
it is only one of several possible stress distributions that depend on the direction of the occlusal force and the relative location of the section in question .  
( 2 ) internal forces (  and  ) and moment (  ) are necessary for the equilibrium of all mandibles , intact or fixed .   beyond sharing  ,  , and  , there are no valid comparisons between intact , plated , screwed , and so forth mandibles .  
( 3 ) the stress distribution within an intact mandible should not be extrapolated to a plated , or otherwise stabilized , mandible .
 
 objectives . the purpose of this analytical study was to examine and critique the engineering foundations of 
 commonly accepted biomechanical principles of mandible fracture repair . 
 materials and methods . 
 basic principles of static equilibrium were applied to intact and plated mandibles , but instead of the traditional lever forces , the mandibles were subjected to more realistic occlusal forces . 
 results . 
 these loading conditions produced stress distributions within the intact mandible that were very different and more complex than the customary lever - based gradient . 
 the analyses also demonstrated the entirely different mechanical environments within intact and plated mandibles . 
 conclusions . because the loading and geometry of the lever - idealized mandible is incomplete , the associated widely accepted bone stress distribution ( tension on top and compression on the bottom ) should not be assumed . 
 furthermore , the stress gradients within the bone of an intact mandible should not be extrapolated to the mechanical environment within the plated regions of a fractured mandible . 
1. Introduction
2. Materials and Methods
3. Results
4. Discussion