ELSEVIER Injury, Int. J. Care Injured 3 1 (2000) S-B 14-S-B23

Stiffness - an unknown world of mechanical science?

F. Baumgart

A0 Technical Commission, Clavadelerstrasse, CH-7270 Davos Platz

Summary’

“Stiffness” is a term used to describe the force needed to achieve a certain deformation of a structure. In the biomechanical world, several different definitions of stiffness are used, but not all of them are explained ade- quately to those readers who are less familiar with bio- mechanical terminology

This paper gives examples for specific definitions which are based on the basic definition of stiffness of a loaded structure

“Stiffness” = “Load“ divided by “Deformation“,

a definition which automatically includes that the “Deformation” is created by the “Load” addressed in the formula.

There is an infinite number of possible configurations of “Load” (forces, moments, stresses, arbitrary groups of forces etc.) acting on a structure and there is also an infinite number of possible points in the structure, where the deformation (displacement, strain, angles, radii, curvature, etc.) can be measured. Therefore, the term “StiffMess” of a structure always requires an exact description of the load configuration and the exact localiza- tion and kind of deformation measured. Otherwise, the mea- sured or calculated values cannot be compared with results from other authors.

An external fixator reacts to different functional forces (axial force, bending moment, torque) with different

1 Abstracts in German, French, Italian, Spanish, Japanese and Russian are printed at the end of this supplement.

2 Stress is defined as force divided by the area on which the force acts.

deformations (axial displacement, several angles, strain etc.). Depending on the selection of the deformations, several stiffness values can be determined. A definition of stiffness as a constant property of a structure only makes sense for structures consisting entirely of linear elastic materials. This paper supports the understand- ing of different stiffness definitions in the literature and provides a simple rule for the control of the stiffness definition applied.

Keywords: torsional stiffness, bending stiffness, material stiffness, structural stiffness, stiffness matrix, load configuration

Injury 2000, Vol. 31, Suppl. 2

Introduction

For the explanation of a well known mechanical phe- nomenon such as “stiffness” some preliminary defini- tions are needed.

If a mechanical structure becomes loaded, a defor- mation occurs.

Here we investigate the phenomenon as seen in a mechanical structure made from an elastic material. In the case of non-elastic materials, the definition of stiff- ness becomes very difficult and unusual.

We understand the terms in this paper as follows:

Y%ucture”

A structure can be a rod, a spring, a plate, a shell or a similar geometrical body consisting of a solid material.

0020-I 383/00/$ - see front matter 0 2000 Published by Elsevier Science Ltd. All rights reserved

Baumgart: Stifiess in mechanical science

“Load”

A load can be a force, a moment, a stress2 or a combina- tion of some of these physical variables acting on the structure.

“Deformation”

Deformation means that the actual geometrical configur- ation of the elastic structure is different from the original “unloaded” reference configuration. A deformation is always a comparison of two different configurations of a structure. The measure of a deformation can be a strains, a displacement, an angle or a modification of these variables. The term “configuration” means that each material particle of the body or structure can be defined by a unique set of coordinates in space (e.g. x,y,z or similar).

“Elasticity”, “elastic material”

Elasticity is a material property which means that a deformed elastic material springs back into its original configuration when the load is removed.

Another definition (which is equivalent to this) means that the internal stress in an elastic material depends only on the strain but not on other deformational vari- ables like strain rate, frequency, or time.

“Stiffness”

After clarification of the above-mentioned definitions, the stifiess ofa structure can be defined as

Stiffness = Load

Deformation5

From this definition the following becomes clear: For an elastic body, the stiffness of a structure is a time

independent value, in other words, the load and defor- mation of the structure have identical time dependency.

Stiffness is not a fixed measure of a structure but is dependent on position and kind of Zoad and on location and kind of deformation.

Therefore: Several definitions of “StiffMess“ exist.

3 The most common definition of strain (the so-called engineering strain) is the difference between the length of a material line in the deformed configuration and the original length in the undeformed configuration, divided by the original length. In other words: The increase in length divided by the original length. There are other definitions in use too, but these are rarely used in the literature.

Injury 2000, Vol. 31, Suppl. 2

S-B15

Basic Examples

The following examples of the different definitions of stiffness in common use (Figs 1-12) are explained in the text under the figures.

Fig. 1: A rod of the length 4 the cross-sectional area A, and the modulus of elasticity E is loaded axially by a force F, the load. The relative axial displacement u of the two ends of the rod is the deformation. In this case, the Longitudinal Stiffness is defined as

s =EA Cl* e

and the basic relation is F=&=S U e

ax

[Comgmasiooofarod:J

F

Fig. 2: Tension and compression are two states of the same load configuration. A force acts on a rod coaxial with the axis of the rod. If the vector arrow of the force at the end of the rod points away from the end surface it is called “tension”, if the arrow points against the end surface it is called “compression”. The left part of Fig. 2 shows a rod under compression. The force F (“load”) causes a displacement II (“deformation”) in the same direction as the load acts. We can use this as a model for the investigation of the tibia under axial load. We apparently have the same situation as in the rod on the left side. This is true if we just ask for the stiff- ness of the system. We can define a stiffness constant S, as the result of the quotient load /deformation, possibly received from

test results. This is possible and permissible. However, calcu- lation of this stiffness must allow for the fact that the cross- section of the tibia is not constant along the axis and the mod- ulus of elasticity also varies in the whole material volume of the bone. Anatomical variations in these properties will influ- ence the stiffness of the system. We additionally neglect the viscoelastic properties of the bone, this means that forces may not only depend on the deforma- tion, but also on the speed of its application. Published test results have shown that this effect in the domain of functional loading variation is small and can be neglected with sufficient accuracy. Fig. 3: An elastic spring, here shown as a helical spring, is the

most common representation of the mechanical term “stiff- mss”. The basic relation between the spring force F and the spring elongation x is

F = cx

where the spring constant c represents the spring stiffness which can be determined according to our fundamental def- inition as

Fig. 4: A spring model (left) of axial loading in part of the

L

F (1)

c =- x

human spine (right) is an extreme simplification of the com- plex real situation in the biomechanical structure of the spine. The axial displacement (“deformation”) x caused by the axial

S-B16

force (“load”) F in a mechanical test is the sum of the local dis- placement differences in the vertebrae and in the interverte- bral discs. All the biological parts of the spine are heterogeneous, have complex shapes, and show more or less anisotropic behaviour. Despite these complex structures a spring constant c can be determined by mechanical testing which allows for the deter- mination of axial deformation caused by axial forces, or for the estimation of the axial forces created by a certain applied axial displacement. The spring constant describes the stiffness of the spine model against axial forces.

It goes without saying that the necessarv assumntion is as alwavs nure elastic behaviour of all Darts of the structure!

Tests at different loading speeds may be needed in order to gather information on the magnitude of hidden viscoelastic effects. This allows an assessment on the accuracy of the assumption of a linear elastic material. This does not mean that all parts of the biological structure must have the same modulus of elasticity. It is also clear that the instability of the spine under compres- sion must be excluded.

Fig. 5: Material stiffness4 is defined by a linear relationship between the normal stress o acting on the surface of a parti- cle and the strain E measured in the same direction as the stress acts. Normally, these variables are determined by a uniaxial tensile or compresssion test.

The “sti@ess” of the material is then defined by the relation- ship

E ~2 E

where the Modulus of elasticity E (YOUNGS modulus) func- tions as the “Material stiffness”. This law is well known as HOOKE’S law of elasticity.

4 of a homogeneous and isotropic material.

Baumgart: Stifiess in mechanical science S-B17

Uaiform saws in the apancumris m. bicipitia bra&ii (t&m. appmximatcly)

r

Fig. 6: Anexample of (approximately) pure tension is the stress transmitted by a tendon to the bone. The tendon axis defines the direction of the axial force. This force divided by the cross- sectional area of the tendon results in almost uniformly dis- tributed axial stress cs (“load”), and uniform strain E (“defor- mation”) of the tendon tissue. The factor of proportionality between the force and the displacement of the elastic tendon material is the modulus of Elasticity E , the “material stiffness”

,R / deformed configuration

Fig. 7: A straight beam is loaded by a pure bending moment M and the deformational response is the radius R. The com- mon definition of the beam stiffness uses the reciprocal value of R,

1 the Curvature K = F

as the deformation parameter.

The basic relationship of the beam theory for bending, which is the most important law used in civil engineering in the analysis for all houses, bridges etc., can be formulated as

M = EJK = EJ f

introducing the Bending stifiess EJ.

This value can easily be determined by a four point bending test, measuring the radius R of the deformed beam.

This definition has the advantage that the length f of the beam in a pure bending mode does not show up, because it is super-

,, deformed contiguratm

Fig. 8: Sometimes another definition can be found in the lit- erature:

Bending moment M Bending angle a “Bending stiffness” E J -

I It is obvious that the definition of the angle a as the deforma- tion introduces the length 1 of the beam into the stiffness defi- nition which has to be compensated by multiplying the cur- vature by the length f Seemingly the length f has an influence on the intrinsic beam stiffness, but in reality it has not. A person bending a longer and a shorter, but sufficiently long rod by hand to the same radius R needs the same bending moment for both tests and he will not feel a difference. This definition has been introduced in civil engineering to facilitate the analysis of structures such as frames and trusses consisting of beams of certain lengths. It has specific advan- tages if the bending moment is not constant over the length of the beam. In such cases, the deformed centre line of the beam is not a circle.

undeformed stsge

twist angle

I

Deformation: Twist: D = $1 torque

Fig. 9: The definition of stiffness for a rod under pure torsional load by a torque Mt is completely analogue to the definition

Injury 2000, Vol. 31, Suppl. 2

of stiffness for pure bending. The basic equation for the load/deformation relationship is here

M, = GJf D = GJt 3

where M, is the Torque (load) and D = ;

is the Twist or torsional angle per unit length (deformation).

The angle 0 is the torsional angle, that means it is the relative rotation of the two end faces of the rod.

Following this we have for the torsional stijfness

G is the shear modulus of the elastic material.

Jt is the torsional moment of inertia of the cross-sectional area of the rod. It depends on the shape and main dimensions of the cross-section. For circular cross-sections, it is identical with the polar moment of inertia, which is twice the value of the moment of inertia ] of a circular cross-section [l]. We see here that this measure of stiffness is independent of the length of the rod. It depends only on the shear modulus (mate- rial property) and on the torsional moment of inertia.

undeformed stage

twist angle

Deformation: Twist angle: Cl

Fig. 10: If we use the torsional angle 8 itself as the deforma- tion variable, the definition of torsional stiffness changes to

s,, = y

In this case, the basic relation for rods under torque load reads

Mt=p e=st2e

I

* 1 . . 1 . I

undeformed

stage *

Fig. 11: Hooke’s law for a material particle under pure she?r stress z is an example of another basic linear constitutive mate- rial equation (see also Fig. 5). The shear stress z depends lin-

S-B18

early on the shear strain y. The material constant is the Shear Modulus G (“Material shear stiffness”).

The material equation reads

G+

'LT - +

Unloaded Deformed Resultant Local intervertebral disc shear shear disc force T stress T

Fig. 12: An example of shear loading is the transmission of A-P forces through the human spine of a standing individual. The intervertebral discs are the less stiff parts in the chain of the human vertebrae, they show the greatest local deforma- tion. In this specific case of loading, the A-P force T results in the distributed shear stress z (“load”) over the end surface of the disc. This causes tilting of all vertical material “fibres” in the disc, it can be described by an angulation y , the shear strain (“deformation”). The factor of proportionality between the two variables is the shear modulus G of the disc material, the mate- rial stiffness against shear stresses. It has to be commented on here that the true shear strain is the trigonometric sinus of the angle between two originally per- pendicular material lines. But in the case of real structural materials (for instance, cortex), the angular changes under functional loading below the failure levels are very small. Therefore, the trigonometric sinus and also tangents can be approximately replaced by the angle itself. This is a common assumption in engineering calculations on real structures and the results show an embarrassing accuracy.

These definitions provide only an incomplete overview. It is obvious that there are some which are not unique (bending stiffness, torsional stiffness). They depend on the definition of the load configuration and on the selected deformation. Furthermore, in most cases the direction of the deformation used is identical to the direction of the “load” used. The following chapter deals with different directions and the more common description of stiffness used in today’s structural analysis methods (Finite Element Method, FEM).

It is also possible to use a load configuration consist- ing of more than one force (e.g. two parallel forces or other combinations of forces and moments). Then “load” means a scale factor to this “unit” configuration.

Baumgart: Stiffness in mechanical science S-B19

Expanded explanation

The material of the structure is still assumed to be lin- early elastic.

This means the stress in the structure depends linearly on the strain. Also all possible deformations depend then linearly on the “load” as long as they are “small” in comparison to characteristic dimensions of the struc- ture or to the dimensionless unit 1.

Comment: This definition is not sufficient in general because a material can be found to be “isotropic” by this test, but it is only isotropic with reference to the axis of this cylinder, but not with reference to other differently oriented axes. We call this property “transversally isotropic with respect to the axis of the cylinder.” But for a first explanation of the anisotropic effect, this exam- ple may be useful and impressive.

Therefore, the stifness values defined above are always constant and independent of load or deforma- tional variables, and as mentioned before independent of the time.

The linear relationship between load and deforma- tion is valid as long as the load configuration remains unchanged by the deformation (Example: In the case of HERTZ’ contact problems between two elastic bodies, the size of the contact area changes due to elastic defor- mation. This leads to a nonlinear relationship between load and deformation despite the fact that the material is linearly elastic! ).

Additionally, we call the behaviour of a material par- ticle “isotropic” if the deformational response of the par- ticle to a stress is the same in all directions.

Example: A circular sheet of metal is loaded by a normal stress parallel to the axis of the cylinder, that means pure compression.

We can measure the radial expansion (displacement) of the cylindrical sheet on two arbitrary, different points of the circumference. If these two values are always the same, we call the behaviour of the material isotropic, if they are different we call it anisotropic (Fig. 13).

Fig. 14: In order to visualize the terms “stress”, “strain” and “anisotropy” applied to biomechanical problems, we must look into the details of the diaphysis of a tibia1 bone A). It is well known that the cortex has an orientation in its structure: Haversian canals helically covered by collagen-hydroxy- apatite microstructures are oriented in a longitudinal direc- tion, other structures show a circumferential orientation, and finally the radial direction is different from the other two. Without better knowledge, we may assume that these three local axes are the main axes of the anisotropy of the bone. This means: Oriented physical properties measured in these three directions would show three different values.

identical loading by normal stress

Anisotropic material LJV*Llx

Deformation behavior is different!

Fig. 13: Isotropic and anisotropic behaviour of materials.

Injury 2000, Vol. 31, Suppl. 2

If we cut a prismatic longitudinal piece of bone out of the tibia B), we have a piece of a truly so-called orthotropic material in front of us. If we cut this to an appropriate length C), compa- rable with the other two dimensions of the prismatic bar, we have found a representative piece of bone showing locally anisotropic behaviour. We can assume that this piece represents the basic local struc- ture of the bone. The whole bone can be puzzled together from such pieces. One piece may show different properties com- pared to others, but the general structure remains the same for all pieces. Only the values of their typical physical properties change from piece to piece. We will now use this typical piece C) to demonstrate the mechanical terminology of the local stiffness properties. The local stiffness parameters determine the overall stiffness con- stants of a structure, even a biological one.

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Example: A quadratic disc of fibre reinforced material with fibre orientation in the x-direction is loaded per- pendicular to the sheet plane (z direction) by a force F. Then the sheet shows different displacements u, und uY in the x and y directions. Its behaviour is anisotropic. In this special case the material is called orthotropic, which is an abbreviation of the term “orthogonal anisotropic” (Fig. 15).

h

Fig. 15: Stiffness coefficients.

If we use the displacement u, for the definition of stiff- ness, we get a certain value of stiffness. If we use uY for the definition of stiffness, the value is different. There- fore, we have to distinguish two different coeficients of stifi ness.

If we refer to an anisotropic (but elastic) material, the definition of stiffness in the above-mentioned sense still holds but becomes more common and difficult because we also have to take into consideration the orientation of the material axes relative to the load configuration that means load axes. We need stiffness coeficients (not only one modulus of elasticity) to distinguish the material axes orientation relative to the load configuration.

In the case of forces and moments acting on structures or bars (trusses, frameworks), this new aspect causes no differences in general.

The stiffness matrix

The most common case of “load” acting on a particle of a material is the stress state. In general, the three-dimen- sional stress state consists of 6 different components of stress: - three normal stress components acting perpendicular to

three different planes. The normal stress always acts parallel to that coordinate axis which is perpendicu- lar to the reference plane and

- three shear stress components acting in three different planes. In each plane two shear stress components act paral-

lel to the two in-plane coordinate axes (Fig. 16).

Txx CT -v = Qx

6 independent stress components

Fig. 16: Stress state on a rectangular particle.

\r Normal stress CJ

L Shear stress T

Fig. 17: Our typical example of a bone piece can be shown by orientation. The axes x,y,z are chosen parallel to the main orthotropic axes of the bone as described above. The physical loading of such a piece, cut (in imagination) from a real bone, is performed by the stress components acting on the surfaces of the piece. There are only two different types of stress: The normal stress o (red) always and only acts pernendicular to the addressed surface. The shear stress r (pink) always and only acts b the addressed surface. Due to the fact that a surface is a 2-D geometrical object, a shear stress acting in a plane always has two differ- ent components, as shown in the picture. The model has six plane surfaces. Three stress components act on each surface. Normally, we would assume that there are 18 independent stress components. However, due to the balance equations for the forces (equilibrium conditions), the three stress components on opposite surfaces must be identical. We now only have 9 remaining different components. But due to the balance equations for the moments, we can conclude that the shear stresses acting on two orthogonal surfaces but in the same plane must be the same: e.g. ZZY = ZYZ This pre- vents rotation of the piece in the three possible planes. We conclude: The whole stress state around our representa- tive piece of bone is determined by 6 independent stress com- ponents: 3 normal stresses and 3 shear stresses.

Baumgart: Stiffness in mechanical science S-B21

The use of strain components is the most common three-dimensional description of local deformation. A complete analogy to the stress state exists:

Example: The elongation in the z direction is obviously L - c The original length is .!. The axial strain in the z-direction is then:

-

-

There are -- three direct strain components, which describe the rela- tive elongation of a particle in the directions of the three axes of the coordinate system and three shear strain components, which each describe the change of the angles between two edges of the parti- cle. In the unloaded state, the two edges are perpen- dicular if we refer to the standard description (Fig. 18).

E = L-l z -

1

this means: Change of length divided by original length. The other component &y can be determined by analogue cal- culation using the lengths in the y-direction. The third axial strain &x must be found analogue, measuring the lengths in x- direction in the x-z-plane or in the x-y-plane. Obviously the strain is a dimensionless geometrical object!

- 3 axial strain components

b _ undeformed 3 shear strain components _ deformed

-angle change

Fig. 18: Strain states of a rectangular particle.

In general, each stress component acting on a parti- cle can cause any of the strain components. That means 6 * 6 = 36 different stifiess coefficients exist.

Strain

L

Shear strain

Y YZ

I

Fig. 20: Many individuals have difficulty in imagining a shear deformation. A shear deformation is very easy to understand. It is the change of the angle between two material lines, which were originally perpendicular to each other. It is evident that we can observe only one of these deforma- tions in each plane. Part A) again shows the y-z-plane. The total angular defor- mation in this plane has two parts (a and p). If we turn the green deformed figure around the point of the angle and bring one branch into coincidence with the direction of the original configuration (black line), part B), then the sums of the two angles become visible and the shear strain yvz is determined.

To achieve a most effective and clear mathematical description matrix calculus is used.

This means we arrange all 6 stress components to a linear “stress vector” s

Fig. 19: This picture shows pure axial strain (tensional strain [extension] or compression strain [shrinkage]). We see the unloaded original configuration and the green deformed configuration of our bone piece. Only the two strain components in the y-z-plane (perpendicular to the x-axis) have been drawn.

Inju y 2000, Vol. 32, Suppl. 2

S-l322

and the 6 strain components to a linear “strain vec- tor” e

r &XX 1 EYY

e= 3 I ! XY &YZ & zx

The relationship between the two vectors for a linear elastic and generally anisotropic material is then

s = Ce

where C is a 6*6 matrix. It is called a Material Stifiess Matrix:

The relationship (2) represents the 6 equations between the 6 stress components and the 6 strain components5.

In the case of an isotropic material, it contains only two independent components, the modulus of elastic- ity E and the shear modulus G (2).

The POISSON number v depends on these two values.

EEEOOO icE-icooo EEEOOO OOOGOO OOOOGO

_O 0 0 0 0 G :

where (2)

,=1-v 1 2G,

1 V

E = l-2v - 2G,

E G = 2(l+v) .

In the case of an orthotropic material (e.g. a fibre-rein- forced material where fibres are oriented in three orthogonal directions), there are 6 independent stiffness coefficients. There are three axes of bilateral symmetry.

A unidirectional fibre-reinforced material is called transversaZZy anisotropic. It has 4 independent stiffness coefficients. It has one axis with rotational symmetry.

Our definition of stiffness as the relation between “load” and “deformation” still holds for this matrix expression. (This means “load” is “stiffness” times “deformation”.)

However, an explicit expression for the stiffness matrix is not available for mathematical reasons (There is no simple calculus for division of matrices).

Theoretically, the stiffness components can be deter- mined by 6 independent tests where always all stress components are set to zero except one. All 6 strain com- ponents can then be measured in each test. From these 36 equations for the 36 stiffness coefficients are avail- able, which is sufficient. (In practice only 21 coefficients are really independent in the case of a generally anisotropic body.)

For materials with special symmetry as mentioned above, less tests are necessary depending on the num- ber of independent coefficients.

Plastic material behaviour

If we increase the stress beyond a certain limit (the yield strength), the material becomes irreversibly changed in structure, i. e., after removal of the load the deformation does not go back to zero, but a so-called plastic defor- mation will remain (Fig. 21).

stress loading-unloading cycle of’ 21 material

A

flow stress

plastic strain elastic slrain

Fig. 21: Elastic - plastic material behaviour.

This means the relationship between stress and strain is non-linear and the definition of stiffness leads to a vari- able value of “stiffness” which depends on the defor-

5 The matrix multiplication on the right side of the equation follows the rule that one selected row of the matrix C is multiplied by the vector e in such a way that the first element of the row is multiplied by the first element of the vector, the second element of the row is multiplied by the second element of the vector and so on. Ail 6 results will then be summed up and deliver that element in the s vector which is located at the position of the selected matrix row.

Baumgart: Stifiess in mechanical science

mation. Therefore, it is in general not reasonable to use the definition “stiffness” in the plastic range of a mater- ial.

This also means: If a non-linear behaviour between load and defor-

mation can be observed in stiffness tests, parts of the structure become plastically deformed. Then the defin- ition of stiffness has to be discussed. Normally the ini- tial elastic range of the structure in the case of low load levels can be used to define the stiffness.

Conclusions

The “science” of stiffness obviously becomes quite difficult if a general view is requested.

But if the definition is consistently applied, difficulties in understanding should not occur. A clear description of the kind and configuration of the load and deformation is always required.

In most cases of use of the term “stifiess”, we should be aware that only a special stiffness coefficient is addressed.

Reference

1. ROARK’s Formulas for Stress and Strain, 6th Ed., New York: Warren C.Young, McGraw-Hill International, 1989.

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injury 2000, Vol. 31, Suppl. 2

ELSEVIER Injury, Int. J. Care Injured 3 I (2000) S-B72-S-B96

A theory is something that nobody believes,

Experimentelle Biomechanik: Teil II Mechanik des Materials

except its author. An experiment is something that eveybody believes,

except its author. (Der National Academy of Sciences,

Washington DC zugeschrieben)

Einleitung: Grundlegende Konzepte und Defi- nitionen der Mechanik

J. Cordey, Dipl. Phys., Dr. SC. A0 Forschungsinstitut, Clavadelerstr., CH-7270 Davos Platz

Zusammenfassung

Diese Einleitung zielt darauf ab, Klinikern die grundle- genden Begriffe der Mechanik von Materialien nahe- zubringen. Was geschieht, wenn ein Knochen (vom Mechanischen her gesehen ein Tr;iger) normalen Bela- stungen ausgesetzt ist: Zentrische Axiallast, Biegung, exzentrische Axiallast, Drehmoment? Wie verformt sich der Knochen? Die grundlegenden Begriffe der Mecha- nik werden unter Verwendung eines Radiergummis unter Last als Anschauungsobjekt prgsentiert, wobei versucht wird, die mathematischen Formeln so weit wie m6glich beizubehalten.

SchliisselwGrter: Mechanik, Knochen, Material, Stress, Dehnung, lineare Biegungstheorie

Steifigkeit - eine unbekannte Welt innerhalb der Mechanik?

F. Baumgart A0 Technische Kommission, Clavadelerstrasse, CH-7270 Davos Platz

Zusammenfassung

&teifigkeit>> ist ein Begriff, der zur Beschreibung der notwendigen Kraft verwendet wird, urn eine bestimmte Verformung einer Struktur zu erreichen. In der Biome-

1 ijbersetzung ins Deutsche: Petra Schwab-Telleria, Pamplona, Spanien

chanik werden mehrere unterschiedliche Definitionen von Steifigkeit verwendet, aber nicht alle liefern ange- messene Erkltirungen fiir einen mit der biomechani- schen Terminologie nicht allzu vertrauten Leser.

Dieser Artikel stellt Beispiele fiir spezifische Defini- tionen vor, die sich auf die Basisdefinition der Steifig- keit einer belasteten Struktur griinden:

&teifigkeib, = (<Last), geteilt durch c<Verformung)).

Diese Definition beinhaltet automatisch, dass die (<Verformung), durch die in der Formel genannte crLastj> verursacht wird.

Es gibt unendlich viele Konfigurationsm6glichkeiten fiir eine auf eine Struktur einwirkende c(Last), (Kr;ifte, Momente, Spannungen, beliebige Krgftegruppen usw.), und es gibt ebenfalls unendlich viele Punkte an der Struktur, an denen die Verformung (Verschiebung, Deh- nung, Winkel, Radien, Biegung usw.) gemessen werden kann. Daher erfordert der Terminus &teifigkeib) einer Struktur immer eine exakte Beschreibung der Lastkon- figuration sowie die exakte Lokalisierung und Art der gemessenen Verformung. Sonst kijnnen die gemesse- nen oder errechneten Werte nicht mit den Ergebnissen anderer Autoren verglichen werden.

Ein Fixateur externe reagiert auf unterschiedliche funk- tionelle Kr;ifte (Axialkraft, Biegemoment, Drehmoment) mit unterschiedlichen Verformungen (axiale Verschie- bung, verschiedene Winkel, Dehnung usw.). In Abhgn- gigkeit von der Auswahl der Verformungen kiinnen v61- lig verschiedene Steifigkeitswerte ermittelt werden.

Eine Definition der Steifigkeit als konstante Eigen- schaft einer Struktur ist nur fiir solche Strukturen sinn- ~011, die ganz aus linear-elastischem Material bestehen.

Dieser Artikel illustriert die Interpretation von ver- schiedenen speziellen Steifigkeitsdefinitionen in der Literatur und verwendet eine einfache Regel zur Kon- trolle der angewendeten Steifigkeitsdefinition.

SchliisselwBrter: Torsionssteifigkeit, Biegesteifigkeit, Materialsteifigkeit, strukturelle Steifigkeit, Steifigkeits- matrix, Lastkonfiguration

OOZO-1383/00/$ - see front matter 0 2000 Published by Elsevier Science Ltd. All rights reserved.

S-B76

Une the’orie est une chose b laquelle

La biomkcanique expkrimentale: Deuxi&me partie personne ne croif sauf Ihufeur.

Une expt?ience est une chose b laquelle

tout le monde croit sauf l’auteur. (Attribube ?I 1’AcadCmie nationale

des sciences, Washington DC)

Introduction P la mhcanique: Concepts de base et definitions

J. Cordey, Dr SC Phys Institut de recherche AO, Clavadelerstrasse, CH-7270 Davos Platz

RCsume

Cet expose des concepts de base de la mkanique des materiaux est une introduction pratique pour cliniciens. Que se passe-t-i1 lorsqu’un OS (en mkcanique une poutre) est soumis B des charges standard: charge axiale centree, flexion, charge axiale excentrke, torsion? Com- ment un OS se dkforme-t-il? Les idles de base du genie mkanique sont illustrt+es B partir d’une gomme que l’on soumet B une charge en maintenant autant que possible les formules mathkmatiques.

Mots cl& mkcanique, OS, matkriaux, stress, tension, theorie de la flexion linbaire

La raideur: un monde mCconnu de la science mkanique?

F. Baumgart Commission technique AO, Clavadelerstrasse, CH-7270 Davos Platz

R&urn6

La ccraideur,) correspond & la force necessaire B la defer- mation d’une structure. Si, en biomkanique, plusieurs dkfinitions sont utilikes pour dkcrire la raideur, toutes ne sont pas toujours clairement expliquees, surtout pour des lecteurs peu familiers des termes spkcialisks. Cet

article presente des exemples de definitions spkifiques issues de la dkfinition de base de la raideur d’une struc- ture mise en charge:

(<Raideur,j = <Charge>) / ~<Dkformatiorw,

oh la c<Dkformatiorw est la conskquence de la (Charge,).

11 existe un nombre illimite de configurations pos- sibles de la <Charge>> (forces, moments, tensions, groupes arbitraires de force, etc.) agissant sur une struc- ture et egalement un nombre illimitk de points de la structure oh l’on peut mesurer la dkformation (dkpla- cement, fatigue, angles, rayons, courbure, etc.). En con&quence, appliquk ?I une structure, le terme c<Rai- deur,, doit toujours correspondre a une description exacte de la configuration de charge et prkciser la loca- lisation et le type de deformation. Sans cette prkcision, les valeurs mesurt?es ou calculkes par diffkrents auteurs ne sont pas cornparables.

Un fixateur externe rkagit aux diffkrentes forces fonc- tionnelles (force axiale, moment de flexion, moment de torsion) en imposant differentes d&formations (dkpla- cement axial, plusieurs angles, fatigue, etc). En fonction des deformations choisies, onpeut dkterminer plusieurs valeurs de raideur.

En tant que propri&! d’une structure, une definition de raideur n’a de sens que pour les structures entike- ment formkes de materiaux B elasticit linkaire.

Cet article est une aide B la comp&hension des diffk- rentes dkfinitions de raideur presentes dans la lit&a- ture et propose une regle simple pour valider la dkfini- tion de raideur utilisee.

Mots cl&: raideur, torsion, flexion, raideur des mate- riaux, raideur structurale, matrice de raideur, configu- ration de charge

2 Traiduit de l’anglais par G. G. Pope, Rennes, France

S-B80

Una teoria 12 qualcosa in cui nessuno crede,

Biomeccanica sperimentale: Parte II Meccanica dei material3

eccetto il suo autore. Un esperimento k qualcosa in cui tutti credono,

eccetto il suo autore. (Attribuito all’Accademia Nazionale

delle Scienze, Washington, D.C.)

Introduzione: Concetti di base e definizioni nella meccanica

suna di esse 6 spiegata adeguatamente a quei lettori the non hanno troppa familiarit& con la terminologia bio- meccanica.

J. Cordey, Fisico Quest0 lavoro fornisce degli esempi di definizioni A0 Research Institute, Clavadelerstr., specifiche basate sulla definizione di base della rigi- CH-7270 Davos Platz dezza di una struttura sottoposta a un carico.

Riassunto

Quest’introduzione ha lo scope di esporre ai medici le idee di base della meccanica dei materiali. Che cosa suc- cede quando un osso (da1 punto di vista meccanico una trave) viene sottoposto a carichi standard: carico assiale centrico, eccentrico, torsione? Come si deforma l’osso? Vengono presentate le idee di base dell’ingegneria mec- canica usando una gomma sotto carico come esempio illustrativo, cercando di ricorrere quanto meno possi- bile alle formule matematiche.

Parole chiave: meccanica, osso, materiali, sforzo, deformazione, teoria della flessione lineare

Rigidezza - un mondo sconosciuto della scienza meccanica?

F. Baumgart A0 Technical Commission, Clavadelerstr., CH-7270 Davos Platz

Riassunto

&igidezza,> 12 un termine usato per descrivere la forza necessaria ad ottenere una certa deformazione di una struttura. Nel mondo della biomeccanica, vengono usate parecchie definizioni diverse di rigidezza, manes-

3 Traduzione italiana a cura di Antonio Pace, Milano, Italia

&igidezza>, = cCarico,> diviso per c<Deformazione)),

una definizione the presuppone automaticamente the la c<Deformazione>) 6 creata da1 Xarico,) indicate nella formula.

C’G un numero infinito di possibili configurazioni di Karico>> (forze, momenti, sforzi, gruppi arbitrari di forze, etc.) the agiscono su una struttura e c’i! anche un numero infinito di possibili punti nella struttura, in cui pui, essere misurata la deformazione (spostamento, estensione, angoli, raggi, curvatura, etc.). Pertanto il ter- mine CXigidezzax, di una struttura richiede sempre un’e- satta descrizione della configurazione de1 carico e l’e- satta localizzazione e l’esatta indicazione de1 tipo di deformazione misurato. Altrimenti, i valori misurati o calcolati non possono essere paragonati con i risultati di altri autori.

Un fissatore esterno reagisce a forze funzionali diverse (forza assiale, moment0 flettente, torsione) con deformazioni diverse (spostamento assiale, vari angoli, estensione, etc.). A seconda della scelta della deforma- zioni, possono essere determinati parecchi valori di rigi- dezza.

Una definizione di rigidezza come proprieth costante di una struttura P sensata solo per le strutture the con- sistono interamente di materiali elastici lineari.

Quest0 lavoro ha lo scope di cercare di far compren- dere le diverse definizioni di rigidezza nella letteratura scientifica e fornisce una regola semplice per il control10 della definizione di rigidezza applicata.

Parole chiave: rigidezza alla torsione, rigidezza alla flessione, rigidezza de1 materiale, rigidezza strutturale, matrice di rigidezza, configurazione de1 carico.

S-B84

Urn teori’a es nlgo en lo que nadie tree,

Biomechica experimental: Parte II Mechica de materialesg

salvo su nutor. Un experimento es algo en 20 que todo el mundo tree,

salvo su a&or. (Atribuido a la Academia National

de Ciencias, Washington DC)

Introduccih: Conceptos bhicos y definiciones de mechica.

J. Cordey, Dipl. Fisica, Dr. en Ciencias En este trabajo se ofrece una serie de ejemplos para Instituto de Investigation de AO, Clavadelerstr., definiciones concretas que estan basados en la defini- CH-7270 Davos Platz cion basica de rigidez de una estructura bajo carga.

Resumen

Esta introduccibn pretende explicar 10s conceptos basi- cos de la mecanica de materiales a 10s medicos. iQu6 Swede cuando un hueso (desde el punto de vista mecanico, una viga) es sometido a cargas estandar: carga axial centrica, flexion, carga axial excentrica, moment0 de torsion? iCorn se deforma el hueso? Los conceptos bdsicos de la ingenieria me&mica son pre- sentados utilizando una goma bajo carga coma objet0 ilustrativo, intentando recurrir lo menos posible a las formulas matembticas.

Palabras clave: mecanica, hueso, materiales, esfuerzo, deformation, teoria de la flexion lineal

La rigidez: jun mundo desconocido dentro de la mechnica?

F. Baumgart Comision Tecnica de AO, Clavadelerstrasse, CH-7270 Davos Platz

Resumen

El termino ccrigidez,, se emplea para describir la fuerza necesaria para conseguir una determinada deformation de una estructura. En el mundo biomecanico se utilizan

4 Traducci6n al espafiol realizada por ijbersetzergruppe Ziirich, Suiza, Pace RiquC

varias definiciones distintas de rigidez, pero no todas ellas se han explicado adecuadamente a 10s lectores menos familiarizados con la terminologia biomecdnica.

&igidez,, = Kargax> dividida por c~Deformaciom~,

una definicibn de la que automaticamente se deduce que la c~Deformaci6r-w es creada por la Karga” a la que hate referencia la formula.

Existe unnumero infinito de posibles configuraciones de (Carga” (fuerzas, momentos, esfuerzos, grupos arbi- trarios de fuerzas, etc.) que pueden actuar sobre una estructura y, ademas, existe un numero infinito de posi- bles puntos de la estructura en 10s que la deformation (desplazamiento, alargamiento, bngulos, radios, cur- vatura, etc.) puede ser medida. Por lo tanto, el termino ccrigidez” de una estructura siempre requiere la descrip- cibn exacta de la configuration de fuerzas y la exacta localization y tipo de la deformation medida. Si no se hate asi, 10s valores medidos o calculados no podran ser comparados con 10s de otros autores.

Un fijador externo frente a una serie de fuerzas fun- cionales distintas (fuerza axial, moment0 flector, moment0 de torsion) reacciona con diferentes tipos de deformacibn (desplazamiento axial, distintos angulos, alargamiento, etc.). Dependiendo de la deformacibn, pueden determinarse varios valores de rigidez.

Dar una definition de rigidez coma propiedad con- stante de una estructura solo tiene sentido en el case de estructuras constituidas en su totalidad por materiales linealmente elasticos.

Este trabajo aporta datos para la comprension de dis- tintas definiciones de rigidez que existen en la literatura y ofrece una regla sencilla para controlar la definicidn de rigidez aplicada.

Palabras clave: rigidez torsional, resistencia a la flex- ion, rigidez de1 material, rigidez estructural, matriz de rigidez, configuration de cargas