JUHA HARTIKAINEN∗, KARI KOLARI† AND REIJO KOUHIA∗
∗Department of Structural Engineering and Building Technology Aalto University School of Science and Technology
P.O. Box 12100, FI-00076 Aalto, Finland
e-mail: firstname.lastname@tkk.fi, web page: http://www.tkk.fi/
†VTT Technical Research Center of Finland P.O. Box 1000, FI-02044 VTT, Finland
e-mail: firstname.lastname@vtt.fi, web page: http://www.vtt.fi/
Key words: Constitutive model, Continuum damage mechanics, Viscoplasticity, Dissipation potential, Ductile-to-brittle transition
Summary. In this paper a simple phenomenological model to describe ductile to brittle tran-sition of rate-dependent solids is presented. The model is based on consistent thermodynamic formulation using proper expressions for the Helmholtz free energy and the dissipation poten-tial. In the model the dissipation potential is additively split into damage and visco-plastic parts and the transition behaviour is obtained using a stress dependent damage potential. Damage is described by using a vectorial variable.
1 INTRODUCTION
Most materials exhibit rate-dependent inelastic behaviour. Increasing strain-rate usually increases the yield stress thus enlarging the elastic range. However, the ductility is gradually lost and for some materials there exist a rather sharp transition strain-rate zone after which the material behaviour is completely brittle.
In this paper a simple phenomenological approach to model ductile to brittle transition of rate-dependent solids is presented. It is an extension to the model presented in1,2using vectorial damage variable3. The model is based on consistent thermodynamic formulation using proper expressions for the Helmholtz free energy and dissipation potential. The dissipation potential is additively split into damage and visco-plastic parts and the transition behaviour is obtained using a stress dependent damage potential. The basic features of the model are discussed.
2 THERMODYNAMIC FORMULATION
The constitutive model is derived using a thermodynamic formulation, in which the material behaviour is described completely through the Helmholz free energy and the dissipation potential in terms of the variables of state and dissipation and considering that the Clausius-Duhem inequality is satisfied4.
Juha Hartikainen, Kari Kolari and Reijo Kouhia
The Helmholtz free energy
ψ = ψ(ǫe, D) (1)
is assumed to be a function of the elastic strains, ǫe, and the damage vector D. Assuming small strains, the total strain can be additively decomposed into elastic and inelastic strains ǫi as ǫ= ǫe+ ǫi.
The Clausius-Duhem inequality, in the absence of thermal effects, is formulated as
γ ≥ 0, γ = −ρ ˙ψ + σ : ˙ǫ, (2)
where ρ is the material density. As usual in the solid mechanics, the dissipation potential
ϕ = ϕ(σ, Y) (3)
is expressed in terms of the thermodynamic forces σ and Y dual to the fluxes ˙ǫiand ˙D, respec-tively. The dissipation potential is associated with the power of dissipation, γ, such that
γ = ∂ϕ
∂σ : σ + ∂ϕ
∂Y · Y. (4)
Using definition (4) equation (2)2 and defining that ρ∂ψ/∂D = −Y, result in equation
σ− ρ∂ψ
∂ǫe
: ˙ǫe+
˙ǫi− ∂ϕ
∂σ
: σ +
D˙ − ∂ϕ
∂Y
· Y = 0. (5)
Then, if eq. (5) holds for any evolution of ˙ǫe, σ and Y , inequality (2) is satisfied and the following relevant constitutive relations are obtained:
σ= ρ∂ψ
∂ǫe, ˙ǫi= ∂ϕ
∂σ, D˙ = ∂ϕ
∂Y. (6)
3 PARTICULAR MODEL
In the present formulation the Helmholtz free energy, ψ, is a function depending on the symmetric second order strain tensor ǫe and the damage vector D, the integrity basis thus consists of the following six invariants
I1 = tr ǫe, I2 = 12tr ǫ2e, I3 = 13tr ǫ3e, I4 = kDk, I5= D·ǫe·D, I6= D·ǫ2e·D. (7) A particular expression for the free energy, describing the elastic material behaviour with the directional reduction effect due to damage, is given by3
ρψ = (1 − I4) 12λI12+ 2µI2 + H(σ⊥) λµ
λ + 2µ(I4I12− 2I1I5I4−1+ I52I4−3) + (1 − H(σ⊥))(12λI4I12+ µI52I4−3)
+ µ 2I4I2+ I52I4−3− 2I6I4−1 , (8) where λ and µ are the Lam´e parameters, H is the Heaviside step-function and
σ⊥= λI1+ 2µ ˆD·ǫe· ˆD, and Dˆ = D/I4. (9)
Juha Hartikainen, Kari Kolari and Reijo Kouhia
To model the ductile-to-brittle transition due to increasing strain-rate, the dissipation po-tential is decomposed into the brittle damage part, ϕd, and the ductile viscoplastic part, ϕvp, as
ϕ(σ, Y) = ϕd(Y)ϕtr(σ) + ϕvp(σ), (10)
where the transition function, ϕtr, deals with the change in the mode of deformation when the strain-rate ˙ǫi increases. Applying an overstress type of viscoplasticity5,6,7 and the principle of strain equivalence8,9, the following choices are made to characterize the inelastic material behaviour:
ϕd = 1 2r + 2
Yr
τd(1 − I4)H(ǫ1− ǫtresh) Y · M · Y Yr2
r+1
, (11)
ϕtr = 1 pn
1 τvpη
¯σ (1 − I4)σr
pn
, (12)
ϕvp = 1 p + 1
σr τvp
σ¯ (1 − I4)σr
p+1
, (13)
where parameters τd, r and n are associated with the damage evolution, and parameters τvp and p with the visco-plastic flow. In addition, η denotes the inelastic transition strain-rate. The damage treshold strain is ǫtreshand the largest principal strain is denoted as ǫ1. Direction of the damage vector is defined through the tensor
M= n ⊗ n (14)
where n is the eigenvector of the elastic strain tensor corresponding to the largest principal strain ǫ1 and ⊗ denotes the tensor product. The relaxation times τd and τvp have the dimension of time and the exponents r, p ≥ 0 and n ≥ 1 are dimensionless. ¯σ is a scalar function of stress, e.g.
the effective stress σeff =√
3J2, where J2 is the second invariant of the deviatoric stress. The reference values Yr and σr can be chosen arbitrarily, and they are used to make the expressions dimensionally reasonable. Since only isotropic elasticity is considered, the reference value Yr has been chosen as Yr= σr2/E, where E is the Young’s modulus.
Making use of eqs. (6), choices (8)-(13) yield the desired constitutive equations.
This particular model has the following general properties:
• Elastic stiffness is reduced monotonously due to damage.
• The model does not include any specific yield stress.
• In the absence of damage evolution, the inelastic model behaves under a constant uniaxial strain-rate loading as
σ → (τvp˙ǫ0)1/pσr when t → ∞, where ˙ǫ0 is a prescribed strain-rate;
• In the evolution of damage, the constraint for the damage D = I4 = kDk that D ∈ [0, 1]
is satisfied automatically, since initially D = 0, ˙D≥ 0 and ˙D→ 0 as D → 1;
• The transition function ϕtrdeals with the change in the mode of deformation through the damage evolution such that
ϕtr≥ 0 and ϕtr≈ 0 when k˙ǫik < η and ϕtr> 1 when k˙ǫik > η;
Juha Hartikainen, Kari Kolari and Reijo Kouhia
• Inequality (2) is satisfied a priori for any admissible isothermal process. Moreover, the dissipation potential (10) is a non-convex function with respect to the thermodynamic forces σ and Y.
• The evolution of damage (6)3 with the potential (11) will result in splitting damage in compression, while for tensile loading damage occurs on the plane perpendicular to the tensile stress10.
• The form (8) of the Helmholtz free energy takes into account the directionality of damage.
The crack deactivation criteria is based on the elastic normal stress acting on the damage plane.
REFERENCES
[1] Fortino, S., Hartikainen, J., Kolari, K., Kouhia, R. & Manninen, T. A constitutive model for strain-rate dependent ductile-to brittle-transition. In von Hertzen, R. & Halme, T.
(eds.) The IX Finnish Mechanics Days, 652–662 (Lappeenranta University of Technology, Lappeenranta, 2006).
[2] Askes, H., Hartikainen, J., Kolari, K. & Kouhia, R. Dispersion analysis of a strain-rate dependent ductile-to-brittle transition model. In M¨akinen, R., Neittaanm¨aki, P., Tuovinen, T. & Valpe, K. (eds.) Proceedings of The 10th Finnish Mechanics Days, 478–489 (University of Jyv¨akyl¨a, Jyv¨askyl¨a, 2009).
[3] Kolari, K. Damage mechanics model for brittle failure of transversely isotropic solids - finite element implementation. VTT Publications 628, Espoo (2007).
[4] Fr´emond, M. Non-Smooth Thermomechanics (Springer, Berlin, 2002).
[5] Perzyna, P. Fundamental problems in viscoplasticity, vol. 9 of Advances in Applied Mechan-ics, 243–377 (Academic Press, London, 1966).
[6] Duvault, G. & Lions, L. Inequalities in Mechanics and Physics (Springer, Berlin, 1972).
[7] Ristinmaa, M. & Ottosen, N. Consequences of dynamic yield surface in viscoplasticity.
International Journal of Solids and Structures 37, 4601–4622 (2000).
[8] Lemaitre, J. & Chaboche, J.-L. Mechanics of Solid Materials (Cambridge University Press, 1990).
[9] Lemaitre, J. A Course on Damage Mechanics (Springer-Verlag, Berlin, 1992).
[10] Murakami, S. & Kamiya, K. Constitutive and damage evolution equations of elastic-brittle materials based on irreversible thermodynamics. International Journal of Mechanical Sci-ences 39, 473–486 (1997).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010
ON CRACK PROPAGATION IN RAILS UNDER RCF LOADING CONDITIONS
JIM BROUZOULIS∗
∗Applied Mechanics
Chalmers University of Technology Gothenburg, Sweden
e-mail: jim.brouzoulis@chalmers.se
Key words: Crack propagation, head checks, rolling contact fatigue
Summary. Three crack propagation methods (based on material forces) are compared for two test cases. From this, a crack propagation law for fatigue growth is proposed. Time integration of this propagation law allows for simulation of head check growth. This is exemplified through a 2D example where the propagation of a single surface crack is simulated for various parameters.
1 INTRODUCTION
Rolling Contact Fatigue (RCF) of rails is a major problem worldwide. Common RCF defects that can be observed in rails are tounge lipping, head checks and squats. However, only the development of head checks will be investigated in this paper. Head checks are typically closely spaced cracks which initially grow almost parallel. The cracks are initiated at the rail surface and are common near the gauge corner in curves. Also the surface friction conditions affect the initiation of head checks. Cracks are more easily initiated under dry conditions than in wet.
To properly simulate the propagation of a crack, we need to model how fast and in what direction it grows. A generalized crack driving force (GCDF ), based on the concept of material forces, is used to model the growth of the crack. Results from simulations using three crack propagation methods are evaluated against experimental results. Based on this evaluation, one propagation method is chosen for subsequent studies.
Next, the propagation of a single head check crack in a piece of rail, under realistic RCF loading conditions, is simulated by the use of a 2D model. Results from the simulations are presented and qualitatively compared to field observations.
2 FATIGUE CRACK PROPAGATION
The GCDF (adopted in this study) can be expressed as1 G =
Z
ΩX
−Σ · (ϕ∇X) dVX (1)
where Σ is the Eshelby stress tensor and ϕ is a suitably chosen weight function of unit value at the crack tip. In this manner, the GCDF is the change of rate of mechanical dissipation due to an advancement of the crack tip.
Jim Brouzoulis
Figure 1: 2D problem setup.
Based on the GCDF for the existing crack, we may formulate a propagation law as
˙a = γ < ˙Φ > ∂Φ
∂G (2)
with the constitutive parameter γ and the crack-driving potential Φ. The expression for the potential Φ is assumed as follows:
Φ = |G| − Gcr (3)
where Gcris a parameter that describes the fracture toughness of the material. By this particular choice of Φ, the crack growth is proportional in direction to the GCDF, which has shown to produce results in good agreement with experiments. It may also be noted that the proposed propagation is of a rate independent type.
The propagation law in eq. (2) is expressed in terms of the crack tip velocity. Therefore, by integrating over one load cycle (N ) the crack growth per load cycle can be computed.
da dN =
Z tN+1 tN
˙a dt (4)
From this, the crack growth is then extrapolated a given number of cycles and the mesh is updated accordingly. The procedure is then repeated until the total number of loading cycles has been reached.
3 NUMERICAL EXAMPLE
To get an understanding of the characteristics of the propagation of a single surface crack in a piece of rail material, a simplified 2D-example is investigated, cf. Figure 1. The material is assumed to be linear elastic and in a state of plane strain. The rail is subjected to the loads from a passing bogie (velocity v) with 2 wheelsets. This produces bending stresses in the rail together with normal and traction stresses in the wheel-rail contact. Bending stresses σb is evaluated from the bending moment developed in the rail as the bogie passes. The normal load pN(x, t) is assumed to be given by an elastic Hertzian contact pressure distribution2 (with with
Jim Brouzoulis
[m]
[m]
higher coef.
of friction
Figure 2: Example of simulated crack paths for varying coefficient of friction (µ = 0.2 − 0.6).
2a). Moreover, the traction stress pT(x, t) is obtained from the normal pressure pN(x, t) and the coefficient of friction µ by assuming full slip, i.e pT(x, t) = µpN(x, t). The traction stress pT acts in the direction opposite to the velocity of the wheel. Furthermore, the crack surfaces are assumed smooth (i.e. no friction).
Results from simulations of crack growth for various parameters will be presented and dis-cussed. The studied parameters are initial crack angle ϕ, initial crack length a0 and coefficient of friction µ. In Figure 2, an example of simulated crack paths for varying coefficient of friction can be seen.
REFERENCES
[1] G.A. Maugin, Material forces: Concepts and applications, Appl. Mech. Rev., 48(5), 213–
245, (1995)
[2] K.L. Johnsson, Contact mechanics, Cambridge University Press, (1985).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds) KTH, Stockholm, 2010c