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Proceedings of the SEM Annual Conference June 1-4, 2009 Albuquerque New Mexico USA ©2009 Society for Experimental Mechanics Inc.

Mesh-size Effect Study of Ductile Fracture by Non-local Approach

Yaning Li and Tomasz Wierzbicki Massachusetts Institute of Technology Room 5-218 Cambridge, MA 02139 [email protected]

ABSTRACT In the numerical simulation of ductile fracture, strong mesh size sensitivity can develop where strain to failure increases with finer finite element meshes. In this investigation, it is shown that mesh size effects occur when using semi-coupled plasticity/damage theory [1] and Mohr-Coulomb ductile fracture model [2] to simulate ductile fracture of the plane strain flat grooved tensile specimen using shell element model with element deletion. The mesh size effects are analyzed from the non-local point of view. It is found that the stress and strain fields have high gradients in the localization zone and the continuing application of the classical stress-strain relation in the localization zone is the cause for mesh size effects in Finite Element (FE) simulations. An equivalent element model is developed to calibrate the non-local stress-strain relation for different mesh sizes. An example of how the mesh size effects can be reduced by using the mesh-size-dependent non-local stress-strain is shown. 1. Introduction Common approaches in the open literature for simulating fracture, for example, are to eliminate elements [3][4]or to allow elements to split [5]when a critical strain to failure is achieved. However, a complication arises due to the observed mesh size sensitivity of the strain to failure which generally increases with finer finite element meshes. A straightforward and conventional way of reducing mesh size effects is to introduce the mesh size dependency to the critical strain to fracture. For example, Simonsen and Törnqvist [5] obtained empirical formulas of the meshsize-dependent fracture strains by curve fitting the results from numbers of finite element simulations with different mesh sizes. Due to the empirical nature of this approach, this method is not general and has many limitations; Li [6], and Li and Karr [7], Li, Karr and Wang [8] derived a close form expression of the mesh-size and material dependent strain to failure by using the concept of strain bifurcation; Lee, Wierzbicki and Bao [9] showed that by taking the average value of critical damage (an analogue of the critical strain to failure) through elements can partially reduce the mesh size effects. Mesh size effects appear after the localized necking, which for ductile materials usually occurs before the initiation of fracture. However, this approach of mesh-size dependent strain to failure only aims at reducing the mesh size effects phenomenologically at the onset of fracture and completely ignores the history dependency of mesh size effects before fracture initiation. After the localized necking, the gradients of stress-strain fields increase dramatically in the localization zone. A number of authors [10-14] pointed out that the first order (conventional plasticity) theory is not accurate when the strain gradients are dominant. Either the physical size effects or the mesh size effect is a pathologic drawback of the conventional theories of plasticity of metals and/or amorphous polymers, which have no length scales entering the constitutive law. Physically, this implies that flow stress at any particular point in a continuum is uniquely

related to the strain at that point, which has been shown to be not true in a series of experiments [15-20]. The earliest formulation of strain gradient plasticity in which the strain gradient effects are represented as the even order Laplacians of the strain field was proposed by Aifantis et al. [21-24] In this paper, an equivalent element model is developed to calibrate the non-local stress-strain relation for different mesh sizes. An initial center imperfection is introduced to trigger localization in this equivalent element model. Numerical simulations of this model are processed with the condition of plane strain tension. It is found that mesh-size effects of the equivalent plastic strain are history-dependent. A 3D surface of the equivalent strain in the space of non-dimensional mesh size and loading history is calibrated. As an initial step of the study, the non-local stress-strain curves are obtained by averaging the stress and strain in the equivalent element model for different mesh sizes. By inputting the mesh-size-dependent stress-strain curves with non-local modification obtained from the equivalent element model, the mesh size effects of the load-displacement curves of the tensile flat grooved specimen are substantially reduced.

2. Mohr_Coloumb Fracture model Semi-coupled plasticity/damage theory [1] is used to simulate ductile fracture. The strategy of the semi-coupled method is to check the equivalent plastic strain of each element for every step. When the fracture criteria chosen is satisfied, the element begins experiencing damage induced softening and will lose strength completely and then be deleted from the FE model until the damage reaches certain critical value following certain damage evolution law. This process is called post-failure behavior. The details of this approach are illustrated by Li and Wierzbicki [1]. A modified Mohr-Coloumb fracture criterion is used to find the equivalent strain to fracture as a function of both stress triaxiality and lode angle parameter . In the limiting case of plane stress, the equivalent strain to fracture can be expressed as Eq. 1.

n A 1 C 2 f 2 1 f ( , ) f3 f 1 C1 3 3 C2 1

(1)

where, n is the hardening exponent of the material, A is the amplitude and

f

is the equivalent strain to fracture.

C1 ,C 2 and C 3 are material parameters need to be calibrated from tests. f 1 , f 2 are the functions of triaxiality ,

f 3 is a function of and the calibrated constant C 3 (The reader is referred to Li and Wierzbicki [1], Bai [2]and

Bai and Wierzbicki [25] for all the details of the derivation.) The fracture model including the pressure and Lode angle dependent plasticity was developed by Bai and Wierzbicki [25]. The damage evolution is defined as

D

d f ( , ) 0

(2)

When the D reaches a critical value, the element will be deleted. Several tests of Trip steel thin sheets are performed to determine the parameters A , n , C1 ,C 2 and C 3 . It is found that A=1275.9 MPa, n=0.2655, C1 0.12 , C 2 720 MPa , C 3 1.095 . These parameters are used in this paper. The details of the experimental calibration of these parameters are similar to [2] and [25]. In this study, we focus on the flat grooved tensile specimen of Trip steel. The dimensions of it are shown in Fig. 1. The tensile test is processed on MTS, load-displacement histories during tension are recorded. The strain before

fracture can be obtained by Digital Image Correlation (DIC) or the measurement of thickness reduction, see Eq.3. For isotropic material, we have

f

2

t ln 0 3 tf

(3)

where, t 0 and t f are the initial and final thickness of the flat-grooved part of the specimen. It is been observed that catastrophic failure occurs suddenly and the crack propagation is unstable.

Fig.1. Drawings of the flat grooved tensile specimen of Trip steel 3. Mesh size effects in simulations of ductile fracture The simple tension of the plane strain flat grooved tensile specimen of Trip steep is simulated using shell element models using the semi-coupled method to predict the tensile fracture of it. Modified Mohr-Coloumb fracture criterion is used. Four mesh sizes are used in the shell model of the flat-grooved tensile specimen. Mesh 6, mesh 12, mesh 24 and mesh 48 in Fig.2 represent the FE shell models with 6, 12, 24 and 48 elements across the flat grooved gauge in the loading direction respectively. The length of the gauge section is 4.99mm (see, Fig.1). The corresponding FE meshes are composed of square elements with the sizes correspondingly of 0.8317mm, 0.4158mm, 0.2079mm and 0.1040mm. There are strong mesh size effects of the numerical simulation results including both the scenarios of the crack paths (see both Fig. 2) and the load-displacement histories (see Fig.3). The drop of load-displacement delayed for mesh 6, while it happens earlier for mesh 48 (see Fig.3).

Mesh 6

Mesh 12

Mesh 24

Mesh 48

Fig.2. Scenarios of cracks of FE shell models of the tensile flat-grooved specimen with different mesh sizes

14.00

without mesh size modification

0.6 PEEQ at the center 0.5 0.4 0.3 0.2 0.1

Equivalent strain to fracture from modified M-C criteria Mesh-size-effect zone before fracture Size-effect zone of fracture

Fracture displacement from the experiment

12.00 Force (KN) 10.00 8.00 6.00 4.00 2.00

S4R, Dc=1.2, D0=1, m=1

Mesh 6 Mesh 12 Mesh 24

Mesh 6 Mesh 12 Mesh 24 Mesh 48 Experiment

0.00 0 0.5 1 Displacement (mm) 1.5

0 0 0.5 1 1.5 Displacment (mm) 2

(a) (b) Fig.3. (a) Mesh size effects on load-displacement curves (b) Mesh size effects on the curves of plastic equivalent strain The phenomenon of the catastrophic fracture of the specimen is captured by these FE simulations. In these simulations, the crack propagation is unstable. The crack initiates when the first element is deleted and then propagates immediately through the gauge and leads the sudden drop of the load-displacement curves observed from both experiments and simulations. Initially, the stress and strain fields are almost uniform along the gauge. After the localized necking, stress and strain localizes in the zone with the scale of mesh-size. If the mesh-size-independent strain to fracture is introduced after the localized necking, as in Fig.3b, the mesh-size-dependent simulation results are obtained. Theoretically, the mesh size effects are present after necking. For Transverse Plane Strain (TPS) state, three necking strains can be obtained analytically using Hill's criteria and/or Bressan-Williams criteria.

diffuse necking n

TPS necking

2

(4) (5) (6)

localized necking

n 3 2n

Numerically, due to the existence of the imperfections (either physical or numerical), the real value should less than these analytical values [6-8]. In this study, it is found numerically, that the strain when mesh size cater in is about 0.214 for all mesh sizes (see Fig.3b).

Undeformed configuration 2

y

1.8 1.6

x

Mesh 6 Mesh 12 Mesh 24

1.4 1.2 PEEQ

y y

1 0.8 0.6 0.4 0.2 0 -4 -3 -2 -1 0 Y-Distance (mm) 1 2 3 4

x x

Fig.4. Typical comparison of plastic equivalent strain distribution along the middle axis y at the same displacement for different mesh size In Fig.4, it is shown in the undeformed configuration that the width of the crack is equal to the size of one element. The finer the mesh, the thinner localization zone and therefore thinner crack width is. If we took an element in the center of the specimen localized within the band of the coarse mesh, the strain in this element is uniform. However, if we took an equivalent element of the same size in the center of the model with finer mesh, the strain in this equivalent element is not uniformly distributed any more.

4. Equivalent element model In order to study the mesh size effects after necking we define the equivalent element model, shown in Fig. 5, where l6 , l12 and l24 are the element sizes of mesh 6, mesh 12 and mesh 24, respectively.

l6

Mesh 24

l12

Mesh 12

l 24

Mesh 6

Fig.5. Sketch of the equivalent element model

O

O

(a) Perfect element (b) Equivalent element with imperfection Fig.6. Loading and boundary conditions for the equivalent element model

The element with larger mesh size ( l6 , l12 ) is modeled in the equivalent element with an initial imperfection (1% thickness reduction). Since the strain is localized into a region with the scale of the mesh size in the loading direction, the size of this imperfection located at the center of the equivalent element is equal to l24 to trigger the localization of the same size (see Fig.6). In this study, the boundary condition is set to be plane strain tension, as shown in Fig.6. For comparison, the larger element with no imperfection is set up to represent the elements of different mesh sizes. Take the element of mesh 24 in the equivalent element model as a reference. The nondimentional mesh size in Fig.7 is defined as

l l24

, where l is the arbitrary mesh size.

0.45 0.4 0.35 Equivalent PEEQ 0.3

Fracture initiates

0.25 0.2

0.15 0.1 0.05 0 0

Increasing damage accumalation

Necking occurs

1

2

3

4

5

Nondimentional mesh size

(a) (b) Fig.7. Mesh-size dependent equivalent plastic strain in the equivalent element model Figure 7 shows that the mesh-size effects are history dependent, where at various stages of the damage parameter, the equivalent plastic strain increases. A 3D surface of the average equivalent strain as a function of non-dimensional mesh size and the representative history variables (Reference PEEQ) is shown in Fig.8. The dark blue area of the surface represents the stage before necking, which has no mesh size effects. The mesh size effects are getting stronger after necking.

0.8 0.7 0.6 0.5

PEEQ

0.4 0.3 0.2 0.1 1 0 1 1.5 2 2.5 3 3.5 4 0 0.5 Reference PEEQ

Nondimensional Mesh Size

Fig.8. The 3D surface of mesh size and loading history dependent PEEQ 5. Non-local modification of the simulations From the analysis of the equivalent element model, we can see that compared with the stress-strain fields of the perfect equivalent element (no imperfection), which are always uniform (zero gradient), the stress-strain fields of the imperfect equivalent element shows very large gradients after localization. The discrepancy leads to the difference between the first order approximation (the first term is the local value) of the non-local field variable defined by the non-local hypothesis [24] and the more accurate non-local field variable with terms of higher order Laplacian. The local stress-strain relation is used in the classical continuum plasticity. After localization, non-local stress-strain curves should be used. Following Bazant et.al. [26][27], a weighted average is introduced to define

the measure of the non-local quantities. In the case of a constant weighting function g1 ( x, y ) g 2 ( x, y ) 1 in Eq. 7 and 8, we have

1 g1 (x, y) ( x, y)dxdy SS

i 1

n

i

(7)

n

where, `

' represents the non-local value of the field. i and i are the equivalent strain and stress of each

1 g 2 (x, y) ( x, y)dxdy SS

i 1

n

i

n

(8)

finite element in the equivalent element model. n is the total number of finite elements in the corresponding area S. Non-local stress-strain relation is shown in Fig.9. Compared with the local reference stress-strain relation, which follows power law, the non-local stress-strain is softened due to localization and the softening is mesh-sizedependent.

Fig.9. Mesh-size-dependent non-local stress-strain curves Input the mesh-size dependent non-local stress-strain relation into the shell element model for the flat grooved specimen with different mesh size, and rerun the simulations. The predicted load-displacement curves and displacement-PEEQ curves from these simulations are shown in Fig.10. By comparing with Fig.3., it can be seen that both mesh size effects before and after fracture initiation are reduced. It should be noted that by choosing the weight function carefully, better results can be obtained. This example shows how the non-local stress-strain modification can reduce mesh size effects for particular mesh sizes.

14.00 with softening 12.00 10.00 Force (KN) 8.00 6.00 4.00 2.00 0.00 0 0.5 1 Displacement (mm) 1.5 coarse mesh 5/6mm mid mesh 5/12 mm fine mesh size 5/24 mm Experiment

0.6 PEEQ at the center 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 Displacment (mm) 2

Size-effect zone before fracture

with non-local modification

Mesh 6 Mesh 12 Mesh 24

Size-effect zone of fracture

(a) (b) Fig.10. Simulation results after non-local modification Acknowlegements

The authors are grateful for the financial sponsorship of this work from ONR MURI project through Stanford University to MIT.

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[11] Fleck, N.A. and Hutchinson, J.W., "Strain gradient plasticity: theory and experiments", Acta Metallurgica et Materiallia 42, 475-487. 1994. [12] Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., Mechanism-based strain gradient plasticity I. Thoery. Journal of the Mechanics and Physics of Solids 47, 1239-1263. 1999. [13] Huang, Y., Gao, H., Nix, W.D. and Huntchinson, J.W., Mechanism-based strain gradient plasticity II. Analysis, Journal of the Mechanics and Physics of Solids 48, 99-128. 2000. [14] Qu, S., Huang, Y., Jiang, H. etc., Fracture analysis in the conventional theory of mechanism-based straingradient (CMSG) plasticity, International Journal of Fracture 129, 199-220. 2004. [15] Fleck, N.A. and Hutchinson, J.W., Strain gradient plasticity: theory and experiments. Acta Metallurgica et Materiallia 42, 475-487. 1994. [16] Stolken, J.S., Evans, A.G., A microbend test method for measuring the plasticity length scale. 1998. [17] Lloyd, D.J., Particle reinforced aluminum and magnesium matrix composites. Int. Mater. Rev. 39, 1-23. 1994. [18] Elssner, G., Korn, D., Ruehle, M., The influence of interface impurities on fracture energy of UHV diffusion bonded metal-ceramic bicrystals. Scripta Metall. Mater. 31, 1037-1042. 1994. [19] Nix, W.D., Mechanical properties of thin films. Metall. Trans., 20A, 2217-2245. 1989. [20] Nix, W.D., Elastic and plastic properties of thin films on substrates: nano-indentation techniques. Mat. Sci. and Engr., A234-236, 37-44. 1997. [21] Aifantis, E.C., "On the microstructual origin of certain inelastic models", Tansactions of ASME Journal of Engineering Materials Technology, 106, 326-330. 1984. [22] Aifantis, E.C., "The Physics of Plastic Deformation", International Journal of Plasticity, 3, 211-247. 1987. [23] Zbib, H, Aifantis E.C., On the localization and postlocalization behavior of plastic deformation. Part I. On the initiation of shear bands; Part II. On the evolution and thickness of shear bands. Part III. On the structure and velocity of Portevin-Le Chatelier bands. Research in Mechanics, 261-277, 279-292, 293-305. 1989. [24] Muhlhaus, H.B. and Aifantis, E.C., A Variational Principle for Gradient Plasticity, International Journal of Solids Structures Vol. 28 (7), 845-857. 1991. [25] Bai, Y. and Wierzbicki, T., A new model of metal plasticity and fracture with pressure and Lode dependence, International Journal of Plasticity, 24, 1071-1096. 2008. [26] Bazant, Z.P., Softening stability: Part I-Localization into a planar band, Journal of Applied Mechanics 55, 517522. 1988. [27] Bazant Z. P. and Pijaudier-Cabot, G, Non-local continuum damage, localization instability and convergence, Journal of Applied Mechanics, 55, 287-293. 1988.

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