1. 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

2. Delta method For elastomeric materials proved an important feature - for the multiplicity of extensions 1 ≤ λ ≤ λ m ≈ 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 Δε ij cause the same increase in stress Δσ ij 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: (1) where: N - is the number of calculation’s steps; φ k - set of geometric parameters that do not change β, and change α k at each stage k; φ k (β, α k ) - 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: (2) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

3. 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 : where: G – modulus of rigidity; μ - Poisson's ratio; p i - external load's intensity; s - hydrostatic pressure function; u i - displacement functions; F σ - body surface on which the given external load; V – volume; i, j – coordinate system (3) (4) (5) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

4. Delta method + Ritz method In the k-th stage of the deformed configuration of the body is described by a local coordinate system: In accordance with the rule changes of variables in multiple integrals: (6) (7) Where: D (x k=1q +Σu qk )/D(x k=1q ) - Jacobian for functions (6), which geometrically characterizes the change in volume of the body V k under the transformation (6). 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

5. Delta method + Ritz method 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 V k =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: (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 - Δ 0, 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: (9) where: F σ - body surface on which the given external load; h – elastomeric layer’s thickness; u z – displacement function by axis z (in the direction of the compressive force - P) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

6. Delta method + Ritz method To the volume’s change can be written: (10) Where: Δ* - shock absorber’s settlement by the weak compressibility of the elastomer, from (9) and (10) is calculated as: 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: (11) For the total shock absorber’s settlement, for the k-th step of loading: (12) 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) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

7. Example Fig.1. Solid conical shock absorber. 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

8. Example For 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: where: (14) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

9. Example Function (14) satisfy the elastomer’s incompressibility: (15) and geometric boundary conditions: (16) For the first stage (k = 1), still undeformed shock absorber for the potential energy U of (4 and 14) We obtain: From the potential’s energy minimum condition (17) for settlement Δk for small strains for the characterization of type “force - settlement” we get: (17) (18) 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

10. Example Accepted that, at each step of loading shock’s settlement are identical, that are: (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: (20) where: 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

11. Example After integration of (20) for the force P, which deforms the shock absorber to the middle deformation, we obtain: 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 s k = P k /GF and from (12) shock absorber’s settlement Δк* by the elastomer’s weak compressibility will be: where: 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania (22) (21)

12. 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: where: 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. 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania (23)

13. 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. 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania

14. 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». 16th International Conference Mechanika. 2011 April 7, 8, 2011 Kaunas, Lithuania