Chapter 3 - Joint and Materials Mechanics Artificial hips Shoe impact tester

Joint Motion Anatomical Position Planes –Sagittal –Frontal –Transverse

Mobility and ROM ROM joint & person specific Injuries: excessive ROM Factors affecting ROM: –Shape and geometry of articulating surfaces –Joint capsule and ligaments –Surrounding muscles –Apposition of body parts Joint Stability

Lever Systems Rigid rod fixed at point to which two forces are applied 1 st class 2 nd class 3 rd class Functions –  applied force –  effective speed RF R F FR

Instantaneous Joint Center Caused by asymmetries in the joint motion Basic movements –rotation –sliding –rolling

Moment of Force & Joint Motion Moment = F ·d  Moment = muscular activity, essential for controlling joint motion Theory: actions at joints can be represented by the resultant joint force and the resultant joint moment

Resultant Joint Force vs Bone-on-Bone Forces RJF: Net force across the joint produced by bone, ligaments, muscle etc. Bone-on-Bone: more complex calculation

Material Mechanics Rigid body mechanics: body segments are considered rigid structures (non-deformable) –fixed center of mass –homogeneous material Used to analyze movements Easier to model and provide a reasonable approximation

Deformable solids Segment or tissue analyzed undergoes deformation More complicated analysis and difficult to model

Material Properties Basic properties –Size –Shape –Area –Volume –Mass Derived –Density –Centroid

Stress Stress (  ): internal resistance to an external load –Axial (compressive or tensile)  =F/A –Shear  = F/A (parallel or tangential forces) Units Pascal (Pa) = 1Nm 2 Axial Shear

Strain Change in shape or deformation (  ) Absolute strain Relative strain –  L/L o

Stress &Strain Stress-strain ratio: stiffness or compliance of the material –E =  /  Linear material –Hooke’ law:  = E·  Biological material non- linear due to its tissue fluid component (viscoelastic properties)   A B

Uniaxial Loading Simplest form: forces applied along single line typically the primary axis –compressive –tensile –shear Stress-strain curve –Linear region (  B ) –elastic limit (  C ) –yield point (  D ) –ultimate stress (  E ) –Rupture (  F ) –Energy stored (area)   BB CC DD EE FF

Poisson’s effect When a body experiences an uniaxial load its axial & transverse dimensions will change, v = -(  t /  a ) Force

Multiaxial Loading Deformation in all three directions Net effect of the strains Shear stresses

Bending Long bones: beams Compressive stress: inner portion Tensile stress: outer portion Max stresses near the edges, less near the neutral axis C T axis  x =(M b ·y)/I y axis

Bending Moments Shear stresses max at neutral axis and zero at the surface  = (Q·V)/(I ·b) Q= area moment V= vertical shear force h b Q y

Bending Three point bending –failure at middle –ski boot fracture Four point bending –failure at the weakest point between two inside forces

Bending Cantilever bending Compressive force acting off-center from long axis

Torsion Twisting action applied to a structure Resistance about long axis determined by polar moment of inertia J=[  ·(r 4 o -r 4 i )]/2 Shear stress along the shaft  =(T·r)/J Twist angle:  =(T·l)/(G·J)

Torsion Larger radius of the shaft, greater resistance Stiffer the material harder to deform In addition to shear stress, normal stress (tensile & compressive) are produced in a helical path (spiral fractures) r

Viscoelasticity Provided by the fluid component in biological tissue Resistance to flow Affects stress-strain Increase in strain rate produces-increases stiffness of the material

Viscoelasticity Pure elastic material –strain energy returned –no energy loss Viscoelastic tissues –lose energy due to heat –energy is not returned immediately –Resilient –Dampened Hysteresis: area representing energy lost   Load unload Elastic Non

Viscoelasticity Creep response Stress-relaxation response Effects of strain-rate on stress relaxation Time  creep 

Material Fatigue & Failure Fatigue: repeated loads above a certain threshold Continued loading: failure First cycle effect: shift in mechanical response   1 23n Initial cycle effect

Material failure Distribution of stresses –Discontinuity (stress risers) fractures sites screws osteotendinous junctions Ductile vs Brittle materials Failure theories –maximal normal stress –maximal shear stress –maximal energy distortion

Biomechanical Modeling & Simulation Model: representation of one or more of an object’s or system’s characteristics using mathematical equations Goal –improve understanding of a system Simulation:process of using validated model

Biomechanical Modeling & Simulation Physical model: simulates actual conditions, crash test dummies Mathematical or computer model: conditions are represented using mathematical equations

Why use a model? Easy to duplicate Easy to make change in the system Time Economic factor

How to select a model? What questions is being posed? –Type: molecular, tissue, organ etc. –Deformable or rigid –finite or continuoum –static, quasi static or dynamic –Linear or nonlinear –2D or 3D –Determined or stochastic –kinematics or kinetics –inverse or direct

Model & Simulations Models are simplifications of actual situations Model and simulation are as good as the data use as input Stability of the model (range of values)

Finite-element modeling Structures are represented as simple blocks assembled to form complex geometrical structures Connected at poinst (nodes) forming a mathetical representation of the structure Forces are applied at the structure and stress and strain are predicted Complex and requires a great deal of computing power

Finite modeling

Rheological Models Study of deformation and flow of matter Use to model biological tissue Interrelate stress, strain, and strain rate Three types –Linear spring –dashpot –frictional Linear spring –elastic properties of tissue  

Rheological Models Dashpot –loading response that is strain rate dependent –fluid viscosity (newtonian fluid) –  =  ·  Frictional element Combinations of models Strain rate    