Transactions on Modelling and Simulation vol 9, 1995 WIT Press, ISSN X

Transcription

1 Elastic-plastic model of crack growth under fatigue using the boundary element method M. Scibetta, O. Pensis LTAS Fracture Mechanics, University ofliege, B-4000 Liege, Belgium Abstract Life of mechanic components is generally linked to crack initiation and propagation under fatigue. Linear models are used to determine stress intensity factors used in Paris laws or similar ones. But it is well known that a plastic zone due to the singularity of the elastic solution exist at the crack tip. Residual stresses generated around the crack tip are important parameters to explain the modification of crack growth rate. This phenomena can be directly observed in case of overloading or crack closure. The Boundary Element Method extended to non-linear 2-dimension problems is used according to the fact that plasticity is localised in a very restricted zone. A first approach tends to link residual stresses in the plastic zone to stress intensity factor. A second approach is based on the plastic energy dissipated at the crack tip. Symbols stress a plastic strain epl stress intensity factor K material constant C, m, n, y cylindrical coordinate (r, 0) boundary Y crack length a domain Q number of cycle N kernels U^T^ size of a plastic zone d Hooke matrix H^ plastic yield op* displacement, traction u, t energy W body force f

2 22 Boundary Element Technology 1 Introduction Since 1852, industries and laboratories have studied the fatigue of metals. Indeed, even today most failures in mechanical structure are due to this phenomena. When a structure is loaded by cyclic external forces, a complete fracture may occur, although a single cycle does not result in any damage at all. The fatigue phenomena can be split in two parts : the initiation of a macrocrack and its propagation until complete rupture. From a micro structural point of view, the two phenomenon are due to the propagation of dislocations existing in any imperfect crystal or polycrystal. The accumulation of dislocations creates a macro-crack. This crack propagates creating an accumulation of dislocation at the crack tip. This paper tries to obtain a good way to predict crack growth rate for medium crack propagation rate. But, it is not realistic to model dislocation at a microscopic point of view. So an elastic-plastic theory is used to model irreversibilities at a macroscopic point of view. 2 Non-linearities 2.1 Linear model PI Until now, the most widely used model to predict crack growth rate is based on an elastic analysis. Near the crack tip, stress expressed in local cylindrical coordinate has always the following singular form : Where K is called stress intensity factor and is calculated and tabulated in a lot of different geometries and loading conditions. Then an empirical law gives crack growth rate according to the variation of stress intensity factor. Paris gives a simple but good law for medium crack propagation rate for symmetrical loading (K^^= -K^x)- It is described in the form of a power law : C and m are material constants. ( J =CAK" AK = K_-K_ (2) It is quite amazing to notice that a linear analysis can provide a good approximation of crack growth rate, because crack propagation is physically linked to non-linear irreversible phenomenon. It has been shown using simple analytical models that there is a link between stress intensity factor and the size of the plastic zone.

3 Boundary Element Technology 23 An other interesting parameter is the total energy dissipated in the plastic zone during a complete cycle. This parameter expresses more clearly the link between crack propagation and irreversibilities. re^dtdq (4) 2.2 Drawbacks of a linear theory PI a) During the loading phase, the maximum stress intensity factor is reach and the crack lips are wide open. During the unloading phase, compressive residual stress are created leading to the contact between crack surface. A new parameter has to be introduced to take into account the effect of asymmetrical cycle. A empirical correction is proposed by Erdogan : K = 5 (5) max n is a new material constants that is not always available. This equation is reduced to Paris law for symmetrical loading. It seems clear that the total plastic energy dissipated increases when R ratio tends to 1. This new material constant n seems to be introduced to take into account non-linear behaviour and could probably be determined using a non-linear analyse. b) All equations and conclusions are only valid for steady state cycling. The stress amplitude is constant and the stress intensity factor amplitude increases slowly. Several papers W have shown that a sudden change in stress amplitude can lead to an acceleration or retardation of the propagation rate. For example, an overloading leads to a large compressive residual zone. Until the crack has propagate through this zone, crack growth rate is lower than predicted with a linear model. A new empirical law taking into account the history is introduced to correct the crack growth rate. Where y is a new material constant, d is the actual plastic zone size due to the last cycle, dj is the plastic zone size created during a previous cycle i and Aa; is the crack length from the cycle i to the last cycle.

4 24 Boundary Element Technology 2.3 Non-linear model The goal of this new orientation is to avoid correction to elastic models and new constants to take non-linear effect into account. A kind of Paris law will be used to predict crack growth rate. -C,m (7) In order to calculate C* and mj, numerical tests will be done for a symmetrical loading in steady state condition. A link between stress intensity factor and plastic energy dissipated will be found: K = C, W"' (8) And a simple identification will give the unknown constant as a function of Paris constant : Ci - C C and m, = m m, (9) 3 Extension of the boundary element method to non-linear mechanics PI In order to model correctly non steady state problems, an elastic-plastic model must be used. Moreover, to express correctly asymmetrical loading nonlinearities coming from contact problems must be implemented (a model without friction will be used). A sub domain technique has been chosen to treat different materials and to avoid an ill conditioned system due to cracks^]. For each sub domain : Somigliana equation is used for each unknown of his boundary : Ui(x') + f^(x',x)uj(x) dlxx) = j"u^(x',x) t/x) df(x) + r r (10) fu,j(x',x) fj(x) d#x)-fs^(x',x)e%(x) do(x) With sj = c%.h^ And the derivation of Somigliany is used for each internal unknown : t,(x) df(x) + f,(x o^ o^ (11) The discretisation of this two linear equations gives : 0 (12) "+f':=0 (13) Boundary conditions can be written as a linear or non-linear equation for contact conditions (friction is not taken into account, so non-linear contact equations are independent of time) : Au + Bt + c = Oand f(u,t) = 0 (14) Plastic behaviour is also described by a non-linear equation :

5 Boundary Element Technology 25 e*=g(e,e) (15) Linear equations (12), (13) and (14) are used to express all unknown as a function of ep* and u^ (u^ is the normal displacement of the surfaces which could be in contact), non-linear equations can be written as follow : f(uv) (16) GP'=g(G^U\GV) (17) It is interesting to notice that the system is relatively small. The number of unknown is limited to the number of non-linear equations. These equations can be solved by any good non-linear solver. 4 Applications 4.1 Bimetallic pipe A pipe composed by an inner aluminium and an outer steel pipe is heated at 50 [ C]. The difference of thermal expansion coefficient creates stresses in both pipe. This interesting example has been chosen to test the sub domain method and to verify equations (12) and (13) where plastic strain (ep*) have been substituted by thermal strain (e^). Material constant : Young's modulus, Poisson ration, coefficient of thermal expansion are respectively: Aluminium : E= [MPa] v=0.33 o=23 10-G[ C"1] Steel : E= [MPa] v=0.3 ct=12 86 _^,.., T [mm] 50 [mm] 50 [mm] Figure 1 : Geometry and Von Mises stress Exact solution is compared with boundary element analysis. Different meshes with degree 2 discontinuous elements have been used. - Ml 1 element on the linear boundary of each sub domain -M2 -M3 2 elements on the linear boundary of each sub domain 4 elements on the linear boundary of each sub domain

6 26 Boundary Element Technology ) Figure 2 : Radial and tangential stress along r axis Mean radial and tangential stress are : <(?n>=14,085[mpa] <G00>=52,7789 [MPa]. It allows to calculate mean errors for each mesh. Grr GAA Ml [%] M2 [%] Table 1 : Radial and tangential stress error in percent M3 [%] Crack under tension A cracked aluminium plate is submitted to uniform pressure. This example is still an elastic solution, but it will be soon treated with the nonlinear model [MPa] A A A A A A A A [mm] Y V V V Y v Y 50 [mm] 50 [mm] _ Figure 3 : Geometry and Von Mises stress

7 Boundary Element Technology 27 Stress intensity factors have been computed using the J-integral method PI. This integration has been made around a circle of radius R and for different meshes (Ml 16 elements, M2 32 elements, M3 64 elements). The exact solution has the following expression W ; V = <x = (18) R = 6.25 R=12.5 Ml% M2% Table 2 : Stress intensity factor error in percent M3% Conclusions The tool presented here can already provide good solutions for linear analyses. In a couple of time, it will provide non-linear solutions and a more detailed analysis will be made to identify the parameters used in the new crack growth rate law. This new model for crack propagation will be used to analyse asymmetrical load and non steady state problems. Results obtained by this analyses will probably reduce experimental testing in order to identify parameters used in empirical laws. References 1. Aliabadi M.H. & Brebbia, Advances in boundary element methods for fracture mechanics, Elsevier, London, Becker A. A., The Boundary element method in engineering, McGraw- Hill, London, Klesnil M, Lukas P., Fatigue of metallic materials, Elsevier, Amsterdam, Murakami Y, Stress intensity factors handbook, Pergamon Press, Exeter, Yan AM & Nguyen D.H., Stress intensity factors and crack extension in a cracked pressurised cylinder, Engineering Failure Analysis, Vol 1 No. 4,pp , 1994.