# Non-Hertzian normal contact of elastic body model by finite elements.

1. ELASTIC FOUNDATION MODEL

The profile, therefore, requires the solution of an integral equation for the pressure. The difficulty is avoided if the solids can be modeled by a simple Winkler elastic foundation or 'mattress" rather than an elastic half-space .The model is illustrated in fig.1. The elastic foundation, of depth h, rests on a rigid base and is compressed by a rigid indenter. The profile of the indenter, z(x, y), is taken as the sum of the profiles of the two bodies being modeled:

z(x, y) = [z.sub.1](x, y) + [z.sub.2] (x, y). (1)

There the difficulty of elastic contact stress theory arises because the displacement at any point in the contact surface depends upon the distribution of pressure throughout the whole contact. To find the pressure at any point in the contact of solids of given is no interaction between the springs of the model, shear between adjacent elements of the foundation is ignored. If the penetration at the origin is denoted by [delta], then the normal elastic displacements of the foundation are given by:

[[bar.u].sub.z] (x, y) = [delta] - z(x, y), [delta] > z, (2)

[[bar.u].sub.z] (x, y) = 0, [delta] [less than or equal to] z. (3)

The contact pressure at any point depends only on the displacement at that point, thus

p(x, y) = (K / h) * [[bar.u].sub.z] (x, y), (4)

where K is the elastic modulus of the foundation.

For two bodies of curved profile having relative radii of curvature R' and R", z(x,y) we can write

[[bar.u].sub.z] = [delta] - ([x.sup.2]/2R') - ([y.sup.2]/2R"), (5)

inside the contact area. Since [[bar.u].sub.z] = 0 outside the contact, the boundary is an ellipse of semi-axes a = [(2[delta]R').sup.1/2] and b = [(2[delta]R").sup.1/2] (Johnson 1985).

The contact pressure by (3), is:

P(x, y) = (K * [delta]/h){-([x.sup.2]/[a.sup.2]) - ([y.sup.2]/[b.sup.2])}. (6)

which is paraboloid rather ellipsoidal as given by Hertz theory. By integration the total load is:

P = K x [pi]ab x [delta]/2h. (7)

[FIGURE 1 OMITTED]

In the axes-symmetric case a = b = [(2[delta]R).sup.1/2] and

P = [pi]/4 (Ka/h) [a.sup.3]/R. (8)

For the two-dimensional contact of long cylinders:

[[bar.u.sub.z] = [delta] - [x.sup.2]/2R = ([a.sup.2] - [x.sup.2])/2R, (9)

so that

p(x) = (K/2Rh)([a.sup.2] - [x.sup.2]), (10)

and the load

P = 2/3 (Ka/h) [a.sup.2]/R. (11)

[FIGURE 1 OMITTED]

In the bi-dimensional case (cylinder), K/h=1.8[E.sup.*]/a, and in the axes-symmetric case K/h = 1.7[E.sup.*]/a where [E.sup.*] is:

1/[E.sup.*] = 1 - [v.sup.2.sub.1]/[E.sub.1] + 1 - [v.sup.2]/[E.sup.2]. (12)

Equations (8) and (11) express the relationship between the load and the contact width. Comparing them with the corresponding Hertz equations, agreement can be obtained, if in the axes-symmetric case we chose K/h=1.70[E.sup.*]/a and in the two-dimensional case we choose K/h=1.18[E.sup.*]/a. For K to be material constant it is necessary to maintain geometrical similarity by increasing the depth of foundation h in proportion to the contact width a. Alternatively, thinking of h as fixed requires K to be reduced in inverse proportion to a. It is consequence of the approximate nature of the model that the value of K, required to match the Hertz equation are different for the two configurations. However, if we take K/h=1.35[E.sup.*]/a, the value of a under a given load will nod be in error by more than 7% for either line or point contact.

The compliance of a point contact is not so well modeled. Due to the neglect of surface displacements outside the contact, the foundation model gives [delta] = [a.sup.2] /2R which is half of that given by Hertz. If it were more important in a particular application to model the compliance accurately we should take K/h=0.60[E.sup.*]/a; the contact size a would then be too large by a factor of [square root of 2].

2. PNEUMATIC TYRES. TRANSVERSE TANGENTIAL FORCES FROM SIDESLIP AND SPIN

The lateral deformation of the tyre is characterized by the lateral displacement u of its equatorial line, which is divided into the displacement of the carcass [u.sub.e] and that of the tread [u.sub.t]. Qwing to the internal pressure the carcass is assumed to carry a uniform tension T. This tension resists lateral deflection in the manner of a stretched string. Lateral deflection is also restrained by the walls, which act as a spring foundation of stiffness K per unit length.

The tyre is deflected by a transverse surface traction q(x) exerted in contact region a [less than or equal to] x [less than or equal to] a. The equilibrium equation is:

[K.sub.c][u.sub.c] - T[[partial derivative].sup.2]/[partial derivative][x.sup.2] = q(x) - [K.sub.1][u.sub.1], (13)

where [K.sub.t] is the tread stiffness. The ground is considered rigid ([u.sub.2] = 0) and the motion one dimensional, so that we can drop the suffixes. Equation (13) can then be solved directly throughout in contact region for any assumed pressure distribution. The carcass deflection are clearly not negligible however and it is more realistic to follow von Schilippe (1941) and Temple (1952) who neglected the tread deflection compared with the carcass deflection ([u.sub.t] = 0, u = [u.sub.c]) as show in fig. 3. Equation (13) then becomes:

u - [[lambda].sup.2][d.sup.2]/[dx.sup.2] = q(x)/[K.sub.c], (14)

where the relaxation length [lambda] = [(T/[K.sub.c]).sup.1/2]. Tafing the case of side slip first, the displacement within the contact region is given by

u = [u.sub.1] - [xi] x x, (15)

where [u.sub.t] is the displacement at the leading edge (x = -a).

Outside the contact region q(x) = 0 so that the complementary solution to (15) gives:

u = [u.sub.1] x [e.sup.{{a+x)/[lambda]}], (16)

a head of the contact and

u = [u.sub.2] x [e.sup.{(a + x)/[lambda]}], (17)

at the back of the contact.

The foundation model is easily adapted for tangential loading also to viscous-elastic solids (Guangming, 2005).

A one-dimensional model of the resistance of a type to lateral displacement is shown in fig.2.

3. ELASTIC FOUNDATION MODEL BY FINITE ELEMENT

The process is iterative and every date when a node by the possible zone of contact is make in contact, the matrix of stiffness it is modified corresponding (Johnson, 1985).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The model is presented m fig.3, the unite plane rectangular elements. In fig.3 is presented the variation of contact pressure between the roll and the rule.

For the 19-27 nodes it was introduced the stiffness (springs) of one constant size for beginning about of Ox,Oy, directions, determinate by the measure of pressure of the 19-27 nodes.

4. CONCLUSIONS

The normal elastic contact could be greatly simplified by modeling the elastic bodies by a simple Winkler elastic foundation rather than by elastic half space. The finite element method are one of the best methods to determinations the pressure of contact.

If the pressure is changed the direction and it is negative and in the anterior node, it is positive, than the limit of the contact zone it's in those case two nodes witch interacted.

If the process is repeated from the intermediate nodes, we find the place where the pressure is changing the sign P>0.

In this way the x coordinate of the respective node represents the semi--breath of contact zone. If every nodes where is in contact, the stiffness matrix is differenced and the maximum stiffness of the elements by who we works carrying on (Rush & Rajkumar, 2000).

The dates are: R = 150 mm, D = 300 mm, b = 40 mm, v = 0.3, E = 2.12 x [10.sup.5] Mpa, K = 3 x [10.sup.8] Mpa--the maxim stiffness in this model case and from this case of loads the semi-breath is a = 63 mm, (Enescu, 2000).

5. REFERENCES

Enescu, I. (2000). Aspecte ale mecanicii contactului la rulmenti (Aspects of mechanics contact of bearings), Lux Libris Publishing House, Brasov

Guangming, Z. (2005). Engineering Analysis and Finite Element Methods. College House Enterprises, LLC, ISBN 0-9762413-1-5, Available from: http://www.collegehousebooks.com

Johnson, K.L. (1985). Contact Mechanics, Cambridge University Press, Cambridge

Rush, C. & Rajkumar, R. (2000). Analysis of Cost Estimating Processes Used Within a Concurrent Engineering Environment throughout a Product Life Cycle. 7th ISPE International Conference on Concurent Engineering: Researches and Applications, pp. 58-67, Technomic Inc., Pennsylvania USA, Lyon, France.

The profile, therefore, requires the solution of an integral equation for the pressure. The difficulty is avoided if the solids can be modeled by a simple Winkler elastic foundation or 'mattress" rather than an elastic half-space .The model is illustrated in fig.1. The elastic foundation, of depth h, rests on a rigid base and is compressed by a rigid indenter. The profile of the indenter, z(x, y), is taken as the sum of the profiles of the two bodies being modeled:

z(x, y) = [z.sub.1](x, y) + [z.sub.2] (x, y). (1)

There the difficulty of elastic contact stress theory arises because the displacement at any point in the contact surface depends upon the distribution of pressure throughout the whole contact. To find the pressure at any point in the contact of solids of given is no interaction between the springs of the model, shear between adjacent elements of the foundation is ignored. If the penetration at the origin is denoted by [delta], then the normal elastic displacements of the foundation are given by:

[[bar.u].sub.z] (x, y) = [delta] - z(x, y), [delta] > z, (2)

[[bar.u].sub.z] (x, y) = 0, [delta] [less than or equal to] z. (3)

The contact pressure at any point depends only on the displacement at that point, thus

p(x, y) = (K / h) * [[bar.u].sub.z] (x, y), (4)

where K is the elastic modulus of the foundation.

For two bodies of curved profile having relative radii of curvature R' and R", z(x,y) we can write

[[bar.u].sub.z] = [delta] - ([x.sup.2]/2R') - ([y.sup.2]/2R"), (5)

inside the contact area. Since [[bar.u].sub.z] = 0 outside the contact, the boundary is an ellipse of semi-axes a = [(2[delta]R').sup.1/2] and b = [(2[delta]R").sup.1/2] (Johnson 1985).

The contact pressure by (3), is:

P(x, y) = (K * [delta]/h){-([x.sup.2]/[a.sup.2]) - ([y.sup.2]/[b.sup.2])}. (6)

which is paraboloid rather ellipsoidal as given by Hertz theory. By integration the total load is:

P = K x [pi]ab x [delta]/2h. (7)

[FIGURE 1 OMITTED]

In the axes-symmetric case a = b = [(2[delta]R).sup.1/2] and

P = [pi]/4 (Ka/h) [a.sup.3]/R. (8)

For the two-dimensional contact of long cylinders:

[[bar.u.sub.z] = [delta] - [x.sup.2]/2R = ([a.sup.2] - [x.sup.2])/2R, (9)

so that

p(x) = (K/2Rh)([a.sup.2] - [x.sup.2]), (10)

and the load

P = 2/3 (Ka/h) [a.sup.2]/R. (11)

[FIGURE 1 OMITTED]

In the bi-dimensional case (cylinder), K/h=1.8[E.sup.*]/a, and in the axes-symmetric case K/h = 1.7[E.sup.*]/a where [E.sup.*] is:

1/[E.sup.*] = 1 - [v.sup.2.sub.1]/[E.sub.1] + 1 - [v.sup.2]/[E.sup.2]. (12)

Equations (8) and (11) express the relationship between the load and the contact width. Comparing them with the corresponding Hertz equations, agreement can be obtained, if in the axes-symmetric case we chose K/h=1.70[E.sup.*]/a and in the two-dimensional case we choose K/h=1.18[E.sup.*]/a. For K to be material constant it is necessary to maintain geometrical similarity by increasing the depth of foundation h in proportion to the contact width a. Alternatively, thinking of h as fixed requires K to be reduced in inverse proportion to a. It is consequence of the approximate nature of the model that the value of K, required to match the Hertz equation are different for the two configurations. However, if we take K/h=1.35[E.sup.*]/a, the value of a under a given load will nod be in error by more than 7% for either line or point contact.

The compliance of a point contact is not so well modeled. Due to the neglect of surface displacements outside the contact, the foundation model gives [delta] = [a.sup.2] /2R which is half of that given by Hertz. If it were more important in a particular application to model the compliance accurately we should take K/h=0.60[E.sup.*]/a; the contact size a would then be too large by a factor of [square root of 2].

2. PNEUMATIC TYRES. TRANSVERSE TANGENTIAL FORCES FROM SIDESLIP AND SPIN

The lateral deformation of the tyre is characterized by the lateral displacement u of its equatorial line, which is divided into the displacement of the carcass [u.sub.e] and that of the tread [u.sub.t]. Qwing to the internal pressure the carcass is assumed to carry a uniform tension T. This tension resists lateral deflection in the manner of a stretched string. Lateral deflection is also restrained by the walls, which act as a spring foundation of stiffness K per unit length.

The tyre is deflected by a transverse surface traction q(x) exerted in contact region a [less than or equal to] x [less than or equal to] a. The equilibrium equation is:

[K.sub.c][u.sub.c] - T[[partial derivative].sup.2]/[partial derivative][x.sup.2] = q(x) - [K.sub.1][u.sub.1], (13)

where [K.sub.t] is the tread stiffness. The ground is considered rigid ([u.sub.2] = 0) and the motion one dimensional, so that we can drop the suffixes. Equation (13) can then be solved directly throughout in contact region for any assumed pressure distribution. The carcass deflection are clearly not negligible however and it is more realistic to follow von Schilippe (1941) and Temple (1952) who neglected the tread deflection compared with the carcass deflection ([u.sub.t] = 0, u = [u.sub.c]) as show in fig. 3. Equation (13) then becomes:

u - [[lambda].sup.2][d.sup.2]/[dx.sup.2] = q(x)/[K.sub.c], (14)

where the relaxation length [lambda] = [(T/[K.sub.c]).sup.1/2]. Tafing the case of side slip first, the displacement within the contact region is given by

u = [u.sub.1] - [xi] x x, (15)

where [u.sub.t] is the displacement at the leading edge (x = -a).

Outside the contact region q(x) = 0 so that the complementary solution to (15) gives:

u = [u.sub.1] x [e.sup.{{a+x)/[lambda]}], (16)

a head of the contact and

u = [u.sub.2] x [e.sup.{(a + x)/[lambda]}], (17)

at the back of the contact.

The foundation model is easily adapted for tangential loading also to viscous-elastic solids (Guangming, 2005).

A one-dimensional model of the resistance of a type to lateral displacement is shown in fig.2.

3. ELASTIC FOUNDATION MODEL BY FINITE ELEMENT

The process is iterative and every date when a node by the possible zone of contact is make in contact, the matrix of stiffness it is modified corresponding (Johnson, 1985).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The model is presented m fig.3, the unite plane rectangular elements. In fig.3 is presented the variation of contact pressure between the roll and the rule.

For the 19-27 nodes it was introduced the stiffness (springs) of one constant size for beginning about of Ox,Oy, directions, determinate by the measure of pressure of the 19-27 nodes.

4. CONCLUSIONS

The normal elastic contact could be greatly simplified by modeling the elastic bodies by a simple Winkler elastic foundation rather than by elastic half space. The finite element method are one of the best methods to determinations the pressure of contact.

If the pressure is changed the direction and it is negative and in the anterior node, it is positive, than the limit of the contact zone it's in those case two nodes witch interacted.

If the process is repeated from the intermediate nodes, we find the place where the pressure is changing the sign P>0.

In this way the x coordinate of the respective node represents the semi--breath of contact zone. If every nodes where is in contact, the stiffness matrix is differenced and the maximum stiffness of the elements by who we works carrying on (Rush & Rajkumar, 2000).

The dates are: R = 150 mm, D = 300 mm, b = 40 mm, v = 0.3, E = 2.12 x [10.sup.5] Mpa, K = 3 x [10.sup.8] Mpa--the maxim stiffness in this model case and from this case of loads the semi-breath is a = 63 mm, (Enescu, 2000).

5. REFERENCES

Enescu, I. (2000). Aspecte ale mecanicii contactului la rulmenti (Aspects of mechanics contact of bearings), Lux Libris Publishing House, Brasov

Guangming, Z. (2005). Engineering Analysis and Finite Element Methods. College House Enterprises, LLC, ISBN 0-9762413-1-5, Available from: http://www.collegehousebooks.com

Johnson, K.L. (1985). Contact Mechanics, Cambridge University Press, Cambridge

Rush, C. & Rajkumar, R. (2000). Analysis of Cost Estimating Processes Used Within a Concurrent Engineering Environment throughout a Product Life Cycle. 7th ISPE International Conference on Concurent Engineering: Researches and Applications, pp. 58-67, Technomic Inc., Pennsylvania USA, Lyon, France.

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Author: | Enescu, Ioan; Vlase, Sorin; Lepadatescu, Badea; Purcarea, Ramona; Dumitrascu, Adela |
---|---|

Publication: | Annals of DAAAM & Proceedings |

Date: | Jan 1, 2008 |

Words: | 1508 |

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