|
|
Basic mechanics |
|
Updated 18/06/2005
|
| Behaviour of materials|loading of structures|Material properties|
Newton by William Blake (1757-1827)
The principles of Biomechanics rest on the three laws of motion set by Sir Issac Newton. 1st Law: An object at rest will remain at rest or an object in motion will remain in a straight line with constant velocity if the net force acting in the object is zero. 2nd Law: An object on which force is exerted will move in the direction of net force and the acceleration will be proportional to the net force 3rd Law: Every action has an opposite and equal reaction The quantities described in biomechanics are either Scalar or Vector. A scalar quantity is one that is defined only by its magnitude. Examples include: mass, work. A vector, on the other hand, has magnitude as well as direction, examples include force,velocity etc. Vector is represented by an arrow. Its length represents the magnitude, orientation the line of action and arrowhead the direction and sense of vector.
Force: Force is action of one body on another. It is a vector. The line of action of a muscle is defined as passing through the centre of muscle. Since it is a vector, a force has a magnitude, a direction and a point of application. Force= mass×acceleration (from Newton's 2nd law) kg.m/sec2=kg×m/sec2 This unit of force is Newton (N)
The red line is the force, it has a point of application, A, sense of force is shown by the arrowhead, the length represents the magnitude and the line of action is the straight line forming an angle measured in an anti-clockwise direction , relative to the axis. Newton is usually too small an unit to be considered. Instead, multiples of N are used. KN= 1000 N, MN=106 N, GN=109 N.
Weight of a body is the force due to gravity. Magnitude of this force is equal to the mass of the body times acceleration due to gravity. So, an average person with a mass of 70kg has a weight=70×10=700 N. Here, Newton is also the unit of weight.
This force acts through the centre of gravity which is usually just anterior to S2 vertebra.
The right handed cartesian system is used as the reference to define position and motion of bodies in space .
Translation and rotation around the direction of the arrows are considered as positive motions.
Resultant of forces: The resultant of two forces can be determined by using the parallelogram law. A parallelogram is constructed by accurately drawing the two forces. The diagonal is the resultant.
Here, R is the resultant of forces P and Q
Rectangular components of a force: A force can be resolved into its component forces in the same way as we resolve a resultant of two forces.
Here, we have the force P resolved into two components, acting perpendicular to each other along X and Y axes. In trigonometric terms, they are expressed as: Py= P Sin
Moment : Moment of a force acting on a body is equal to the product of the force times the perpendicular distance between the line of action of the force and the axis of rotation. Moment is essentially the measure of a tendency for a force to rotate it about an axis. There is a reaction force at the centre of rotation to counteract the applied force.
Here, a force P acts on an object at a point which is d distance away from the centre of rotation. This distance is also known as the moment-arm of the force. The moment is negative as it would move the object in a clockwise direction. Under cartesian system the moment is positive if it rotates the body in an anti-clockwise direction and vice versa. The unit of moment in international system is newton-metre.
Couple: If two forces acting on a body separated by a distance from each other have equal magnitude, parallel line of action and opposite sense then they form a couple. It is also commonly known as torque.A couple acting on a body rotates the body.There is no net reaction frorce in the body.
Law of equilibrium: From Newton's first law of motion, we can establish the law of equilibrium. There may be a number of forces acting on a body, if there are no net forces or moments , then the body is in equilibrium. ∑ F=0 ∑ M=0
From Newton's first law of motion, we also develop the concept of Inertia, which is the tendency for a body to resist change in its velocity.
Free body diagram: When we consider a body to determine the forces acting on it, it is called a free body. A body under consideration has a number of external forces acting on it. The free body diagram is a simplified sketch of the body under consideration showing all the external forces acting on it . The FBD also shows the co-ordinate system being used which defines the positive directions. The forces found from FBD are approximate only. Various assumptions are made while constructing a FBD. Muscle forces are considered to act parallel to the long axis of the muscle along the centroid of muscle cross- section. An appropriate free body needs to be constructed when considering forces across any joint. Gravity always acts across a free body. Depending on whether or not there is ground contact, ground reaction force would provide equal and opposite reaction. Muscle forces crossing the free body influence its equilibrium.
Free body diagram of a person standing up from a chair
Friction:
If we consider a body with weight W resting on a flat surface, then due to Newton's 3 rd law of motion, an equal and opposite force R is directed towards the body. If we now provide a force P to move the body and then fail to do so, this is because there is a force F opposing P. This force F is the friction force and is proportional to reaction force R. µ is the co-efficient of friction. Frictional torque is the force due to friction of a bearing. T=µ><r2 Charnley low friction arthroplasty: John Charnley first popularised and succeeded in the concept of Low friction arthroplasty. He realised that using a small diameter head for a hip arthroplasty would reduce the moment arm of friction and frictional torque. But his success is also due to using UHMWPE which has low wear rate.
σ=F/A
Stress can be of different types: tensile, compressive, shear . Tensile stress pulls, compressive stress pushes and shear stress acts parallel to the surface.
Strain: It is a measure of deformation. It is the change in length produced by a force divided by the original length. It does not have any unit. Tensile strain produces elongation and compressive strain produces shortening. Shear strain produces a change in angle. Here force P acting on an object with the black line has changed its angles to the red margin to produce shear strain.
Hooke's Law: In most solid objects, the relation between load and deformation is linear. This is known as Hooke's Law. F= kx f= force, k= constant , related to stiffness of the object, x= deformation This linear relationship is only true for elastic solids. Plastic objects and soft tissues inside the body have a non-linear load-deformation curve. So, when we say an object is
Elastic: it means that when deformation is released, there is no permanent change in the shape of the structure.
So, when an object undergoes plastic deformation, there is permanent and non-recoverable deformation. This brings the concept of ductility.
Ductility: is the measure of the degree of plastic deformation sustained before material failure of the object. Bone has a ductility of 3%, UHMPWE 300% and 316 L stainless steel 30%.
A Brittle material sustains little or no plastic deformation before fracture. PMMA bone cement is very brittle. So, it is obvious that engineering materials have two modes of fracture or failure: ductile and brittle. Ductile materials exhibit a degree of plastic deformation before failure, brittle materials show little or none.
a= highly ductile fracture, necking b= moderately ductile fracture( this is the common type of tensile fracture in ductile materials and is called cup and cone fracture), c= brittle fracture, they break by rapid crack propagation and the surface is almost flat. However, they do have microscopic v shaped chevron markings on their surface.
Elastic Modulus: the stress-strain curve is constant for elastic region . This constant is called Young's Modulus (E). Young's modulus: we have seen that for most objects, load-deformation curve is linear, in other words, relationship between stress and strain is proportional to each other. so,
σ=E
ε σ=
stress,
E=Young's modulus, ε=strain Young's modulus is constant for each material and is a measure of a material's ability to maintain shape under external loading. The higher the moduli, the stiffer the material. The value for titanium is 107 GPa ( Giga Pascal), Steel 207 GPa. Tendon , Cortical bone is 18 GPa, Cancellous Bone , PMMA is 3 GPa and lowest is UHMWPE at 1 GPa.
Inelastic region of stress-strain curve: elastic materials , if subjected to continued stress, loose their elastic property . At this stage, even if the material is unloaded, it would not return to its original shape. This is the plastic region of the stress-strain curve. This knowledge helps us to identify how much can safely be carried by an implant before it is deformed.
Anisotropicity: some materials behave differently at different directions, this is called anisotropic behaviour. Metals are isotropic, bone is anisotropic,as is intervertebral disc.
Viscoelasticity: some materials exhibit properties which change over time, in spite of constant loading, this is Viscoelasticity. Most organic materials like bone, cartilage, tendon, ligaments have viscoelasticity. There are four basic characteristics inherent in viscoelasticity. They are:
Creep: increasing deformation of a material under constant loading.
Stress relaxation: over time stress in a object would reduce inspite of constant deformation. In other words, stress would reduce under constant strain.
Hysteresis curve: these two properties introduce the concept of Hysteresis curve. In an elastic material , the loading and unloading curve follows the same curve. In a visco-elastic material, it is different. The difference between the two represents the energy that is lost during loading.
Loading rate sensitivity: Visco-elastic materials also show loading rate sensitivity which is also present, albeit, less prominently, in non-viscoelastic materials. This is inherent in biologic materials like bone, tendon, ligament,disc etc. At faster loading the material becomes more rigid and needs larger force to failure. The curve also illustrates that at faster loading, the material stores more energy before failure.This gives bone a biologic protection. Bone is 30% stronger for brisk walking than for slow walking.
Fatigue: we suffer from fatigue in life due to the stress of modern life. So, can a structure. We have seen that structures can undergo ductile or brittle fracture depending on their material property. This definition is subject to a static load. Most of the structures in life undergo repetitive and fluctuating loading. Under this circumstance, failure can occur at much lower stress level than the tensile or yield strength for static load. They are the commonest mode of structural failure and show little or no foreboding with prior deformation , even in ductile materials. Cyclic loading causes micro-structural damage to an object that accumulates with each loading cycle. Damage accumulates faster at higher intensities of loading. Stress fracture in bone is an example of fatigue failure. Some of the Femoral stem prosthesis have also had fatigue failures. Fatigue initiates at a point of stress concentration which is called stress raiser. Factors affecting fatigue life include any notch, groove, thread, hole, surface markings etc. They can all act as stress raisers.This is why it is vital while putting joint implants to make sure that there is no inadvertent scratch mark.
Corrosion: Body fluid has high salinity and any metal implanted inside would come into contact with it. So, it is important to ensure that implanted metals maintain their mechanical properties under attack. Metals can undergo electro-chemical attack in contact with body fluids and deteriorate. This mechanism involves oxidation-reduction reactions. Materialls have different tendency to dissolve and this is indicated by their electrode potential. Magnesium is most active , followed by zinc, cast iron, 316 L stainless steel, titanium, gold, platinum. So, from the sole point of view of corrosion resistance, platinum would be the safest metal to implant. The fact that metals have differential tendency to corrode can be utillised to allow preferential corrosion of one to protect the other. This can only be achieved in a controlled environment,ie. larhge anode/small cathode.For ordinary purposes , a design combination of two metal implants invites early failure through corrosion. Corrosion can be of different types, stress, crevice, pitting etc. An important type is fatigue corrosion, where a combination of cyclic fatigue and chemical corrosion can spell an early doom for an implant.
|Loading of structures| Body structure are under considerable load. This loading results in development of substantial stress and strain Depending on the point of load, there is tensile load on one side and compressive load on the other side. Axial load: applied parallel to the longitudinal axis of the structure. This type of load can be either tensile or compressive. Torsional load: also known as torque. It is a rotatory load applied by force couple that tends to twist the structure.
Bending moment (M): If a load is applied to a structure a distance away from the long axis, the effect is to bend the structure. This load creates a bending moment.
Cantilever bending: a special type of bending load where load is applied to a structure at its free end perpendicular to the long axis and the structure is fixed at other end. Maximum bending moment would occur at the fixed end. Best example is a man standing on the end of a diving board.Amount of bending moment varies along the length of the board and is equal to the product of
Three point bending: when load is applied to a structure supported at two ends at any point perpendicular to its long axis.
Maximum bending moment is under the force F. From moment equilibrium , we find that Fa= Fb/(a+b) Fb=Fa/(a+b)
Four point bending: load is applied to a structure in the middle supported at both ends, two equal and parallel forces are applied in the middle.
Bending moment is highest at the point of application of the forces and equal between them, lowest at the two ends.
Neutral axis:A beam subjected to pure bending
will deform with compression on superior aspect and tension on inferior
aspect. At the neutral axis, there is no change.
Bending of a beam: A cantilever beam mounted on a wall in subjected to tensile strain on top and compressive strain on the bottom. There is nil strain on the neutral axis. Greatest strain and stress would occur on the periphery of the beam, or on the point furthest away from the neutral axis. The way to minimise this stress is to put more material on the periphery of the structure. This brings us to the concept of second moment of area.
Second moment of area (I): If a beam is subjected to a force , we need to know how much stress would develop in order to be able to predict how much load it can withstand. Engineer's equation for bending stress in a uniform beam is
σ=My/I M=
bending moment, y=distance from neutral axis, I=second moment of
area
Second moment of area is a measure of the amount of material distributed around the cross section of a beam. This knowledge helps to estimate its bending rigidity. For a rectangular beam, it is dependent on height, for a beam with circular cross-section it is dependent on radius.
For a beam with circular cross-section , the second moment of area is
I=Π/64(di4-d24)
di
is outside diameter d2 is inside diameter
Flexural rigidity of a beam is equal to bending stiffness times the second moment of area. A long thin beam is stronger than a flat one( less height).
Polar moment of inertia (J): polar moment of area comes into play when we consider the torsional rigidity of a structure. It follows the same principle. So, we have more strength if the material is away from the centre. For a hollow circle, J is
J=Πd4/32
Because mass is situated away from the neutral axis, a hollow circular shaft with equal material is stronger than a solid shaft with the same amount of material. On a different line, if we want to replace a solid circular shaft with a hollow shaft with equal stress then by making the shaft hollow, we could save a lot of mass. Reamed hollow nails are stronger than solid unreamed ones. Construction of human bone follows the same principle.
|Material properties|
Metal : pure metals often are quite weak and may have many undesirable properties. In order to avoid this drawback ,metals are mixed together to produce alloys. So, an Alloy is a composite material. Most of orthopaedic materials in use are alloys. Examples include 316 L Stainless steel, Titanium alloy, Cobalt-chrome etc. Stainless steel : alloy of iron, chromium, nickel. 316 L has chromium and is corrosion resistant. This is important as implants inside the body are exposed to the very corrosive influence of saline. It is still rather susceptible to corrosion and has low fatigue strength. Co-Cr: The most useful Co-Cr alloy is MP35N. It has better corrosion , fatigue and strength than 316L steel. It has high abrasion resistance and is a good bearing material. It is the common prosthetic implant. Titanium: the most useful titanium alloy is Ti-6Al-4V. Titanium is extra-ordinarily light, strong, highly ductile, corrosion and wear resistant. Its elastic moduli is closest to cortical bone. Titanium is , however, very notch sensitive. Ceramic: they are compounds between metallic and non-metallic elements. Some of them assume crystalline structure, others can be amorphous. Ceramic is used in orthopaedics as an alternate bearing material because it has low wear rate, is bioenert and is extremely hard. But it is also brittle and fracture is a concern. Its implantation is also technique sensitive.
Polymer: they consist of long chain of carbon atoms to which various other atoms are bonded at the sides. Polymers of orthopaedic interest include UHMWPE and PMMA.UHMWPE is used as a bearing material in joint replacement surgery. It is highly resistant to wear, abrasion, impact and has low co-effecient of friction. UHMWPE is visco-elastic and would undergo creep under constant load. Although the wear rate is low, UHMWPE wear particles can induce significant reaction and newer highly cross-linked variety as well as better sterilisation technique have been developed to reduce wear.
|
|
,
Px= P cos
M=Pd
