calculation of rubber shock absorbers at compression at middle deformations taking into account...

Calculation of rubber shock absorbers at compression at middle deformations taking into account compressibility of elastomeric layer.

V. Gonca, Y. Shvabs Riga Technical University, Institute of Mechanics

16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

Delta methodFor elastomeric materials proved an important feature - for the multiplicity of extensions 1 ≤ λ ≤ λm ≈ 0,5 ÷ 0,6 (middle deformation) there is a linear correspondence between stress σij and multiplicities elongations λ, that is regardless of the loading initial level the same strain increment Δεij cause the same increase in stress Δσij and vice versa. This allows for the initial state to take pre-strained state of the body, for which small additional strain to apply the linear theory. In this case, the middle deformation, using linear physical relations proposed a simple algorithm for calculating - delta method:

N

PP

NP kkkk

,),( (1)

where: N - is the number of calculation’s steps;φk - set of geometric parameters that do not change β, and change αk at each stage k;φk(β, αk) - decision at each stage.

For example, we determine at a middle deformation, shock absorber’s settlement Δ, depending on the force P and already obtained the recurrence relation for the k-th stage:

0 01

),( ddPPPN

kk (2)


Ritz method

This delta method’s scheme for secondary deformities in the calculation for the elastomeric products of the stiffness characteristics (type of “force –settlement”) is most effectively used with the Ritz method for the functional :

F

iiV

iii dFupdVsuJuJU ),()( ,21

ijiji GuJ )(1

2

2,2)1(4

219

1)( ss

FGuJ iiii

where: G – modulus of rigidity;μ - Poisson's ratio;pi - external load's intensity;s - hydrostatic pressure function;ui - displacement functions;Fσ - body surface on which the given external load;V – volume;i, j – coordinate system

(3)

(4)

(5)


Delta method + Ritz method

In the k-th stage of the deformed configuration of the body is described by a local coordinate system:

1

11

k

qkqkkq u

In accordance with the rule changes of variables in multiple integrals:

(6)

kqV

kikiikqkkV

kikiik dVDsuJuJuDdVsuJuJk

)(),()()(),()( 1,21,21

11

(7)

Where: D (xk=1q+Σuqk)/D(xk=1q) - Jacobian for functions (6) , which geometrically characterizes the change in volume of the body Vk under the transformation (6).



For an incompressible material, this Jacobian is equal to unit. Therefore, if the body’s material is considered incompressible, then any k-th step of solving the integration in (3) can be performed on the initial undeformed volume of the body Vk=1, which greatly simplifies obtaining solutions for middle deformations. If at each step in choosing displacement functions satisfy the condition of incompressibility, instead of the functional (3) at the k-th step of the solution should be used functional:

1

110 )(V F

ikikik dFupdVuJU (8)

For shock absorbers under axial compression (even compressive force applied along the axis z) to take into account the weak compressibility of the elastomeric layer on the integral characteristic, of type ”force - settlement”, found in the assumption of incompressibility of the elastomer - Δ0, it is possible to find approximately by two methods. In a first variant of the physical relations between the strains and stresses for the weakly compressible elastomers for changes in the volume we have the relation:

dzdF

z

u

G

PsdVdVu

h V

F

z

VVii

0

11,

1

112

21

1

21

2

3(9)

where: Fσ - body surface on which the given external load; h – elastomeric layer’s thickness;uz – displacement function by axis z (in the direction of the compressive force - P)



To the volume’s change can be written:

FVdVuV

ii *

1,

1

(10)

Where: Δ* - shock absorber’s settlement by the weak compressibility of the elastomer, from (9) and (10) is calculated as:

dzdFz

u

G

P

F

h

P

z

0 2

21

In the second version, if calculating the sediment Δk without compression elastomer, found expression for the hydrostatic pressure sk , then from equation (9) for precipitation Δk* to obtain the expression:

1

1)1(2

)21(3

kV

kkk dVsF

(11)

For the total shock absorber’s settlement, for the k-th step of loading:

(12)

kkk

Summarize the solutions for all steps (k = 1, ... N), replacing the summation by integration in the N → ∞ (if the formula (13) can be written in recursive form), we obtain the desired force-displacement relationship for the middle deformation.

(13)


Example

Fig.1. Solid conical shock absorber.


ExampleFor small deformations, assuming an incompressible elastomer, using the results of work. In cylindrical coordinates (r, Ө, z) the displacement functions, respectively, along the axes r, Ө, z we can write:

)(sin),( tgrzh

nArzrur

00 u

)(sin)(1)(cos

2),( tgrz

h

ntgrtgrz

h

nhAzruz

where:

hA

4

(14)


Example

Function (14) satisfy the elastomer’s incompressibility:

0,

z

u

r

u

r

uu zrr

ii(15)

and geometric boundary conditions:

0))(,())(,( tgrhzrutgRrzru rr

),(,0),( tgrhzrutgrzru zz(16)

For the first stage (k = 1), still undeformed shock absorber for the potential energy U of (4 and 14) We obtain:

Ph

Rtg

h

GRU

kk

242

222

23

1cos8

)(25.2316

From the potential’s energy minimum condition (17) for settlement Δk for small strains for the characterization of type “force - settlement” we get:

(17)

)(cos8)(25.23125.0

4

22230 k

kk tgGR

hP (18)


Example

Accepted that, at each step of loading shock’s settlement are identical, that are:

h

khhk0)1(1

N

N

kk

10,

(19)

Summing all forces Pk and passing in (18) to the limit as N → ∞ and Δ → 0, we find the total force of P, is deforming shock absorber to the middle deformation 1 ≤ λ ≤ 0.5 ÷ 0.6:

1)1(

1

)(cos8)(25.23125.0

1

024

22223 d

tgGRP (20)

where: h

R

hhN

;11 0


Example

After integration of (20) for the force P, which deforms the shock absorber to the middle deformation, we obtain:

1)(cos16

ln)(25.23125.0 24

221223 tgGRP

Additional shock absorber’s settlement Δ* by the weak compressibility of the elastomer can be calculated using (11) and (14). Function (14) describe the displacements for an incompressible material, so and settlement Δ* in (11) will be approximate, since it ignores the real distribution of displacements for the compressed material. In the first approximation, to evaluate the effect of weak compressibility of the elastomer on the draft of the shock absorber can use the simpler formula (12), assuming that the volumetric deformation under consideration conical elastomeric layer under the influence of the total compressive force P can be calculated, if this layer is totally rigid matrix under uniaxial deformation in the absence of friction. In this case we can assume that sk = Pk /GF and from (12) shock absorber’s settlement Δк* by the elastomer’s weak compressibility will be:

)1(2

)21(3

GF

P

hk

k

k

where: 2RF


(22)

(21)

Example

Applying (13), (18) and (22) procedure for the delta method, described above,to find the total force P is deforming shock absorber to the middle deformation subject to elastomer’s weak compressibility of the:

A

AFGP

)21(125.01

ln125.02

12

where:

h

R

h

tgA

;1;ln

)1(0625.0)(25.231

2222

For sufficiently large ratios Δ*/h, and Δ/h calculation should be performed immediately for a compressible material, since the estimate (22) can be a rough approximation.


(23)

Results

Dependence “force - deformation” for Solid conical shock absorber. R = 5 cm, h = 5 cm, β =

20°, G = 120 N/cm2, μ = 0,492

Dependence “force - deformation” for Solid conical shock absorber. R = 10 cm, h = 2 cm, β = 20°, G = 120 N/cm2, μ = 0,492

“1” line - by formula (18), μ = 0.5;“2” line - by formula (21), for middle deformation, μ = 0.5;“3” line - total deformation for the middle deformation, taking into account elastomer’s weak compressibility.


Conclusions

The proposed method allows calculate elastomer shock absorbers at middle deformations, taking into account the weak compressibility of the elastomer. Using the techniques demonstrated by axial compression of conical shock absorber, the results are show with charts. No allowance for the weak compressibility of the elastomer at middle deformation when the elastomeric layer is thin enough, can lead to large errors when calculating this shock absorber deformation (fig.4.). In turn, if the shock absorber is high the effect of weak compressibility of the elastomer is not significant. (fig.3.)

Acknowledgments His work has been supported by the European Social Fund within the project «Support for the implementation of doctoral studies at Riga Technical University».


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