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64, No. l-4, pp. 643-653, 1997 Science Ltd. All rights reserved

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COUPLING OF MICRO- AND MACRO-DAMAGE MODELS FOR THE SIMULATION OF DAMAGE EVOLUTION IN DUCTILE MATERIALS C. Kiinke Institute for Statics and Dynamics, Ruhr-University

Bochum, D-44780 Bochum, Germany

Abstract-This research presents a new simulation concept of damage evolution for metallic materials under large deformations. The complete damage range is subdivided into both the micro- and macro-dama,ge range. The micro-damage phase is described by the Cocks-Ashby void-growth model for isotropic, ductile materials under isothermal conditions. After having reached a critical void-volume fraction, a macro crack is introduced into the model. Applying the finite element method for the numerical formulation, at every incremental macro-crack step the finite element mesh is adapted such that the crack path remains independent of the initial mesh. With this concept the damage evolution from nucleation and growth of micro-voids to initiation of macro cracks and complete failure of the material can be described. CI 1997 Civil-Comp Ltd and Elsevier Science Ltd.

1. INTRODUCTION Since the beginning of this century attempts have been made to describe the damage evolution of processes in which material elements were damaged or even destroyed. Kachanov formulated in 1958 damage evolution. in creep [l]. Griffith studied macro-crack probbms [2]. Without effective numerical tools, both were only able to describe simple problems or use phenomenological laws. Today, complex processes in structures with arbitrary boundary conditions can be simulated numerically. As a result, new material descriptions with theoretically based damage evolution laws have been developed. In continuum damage theories the presence of a number of micro-defects is effectively smeared throughout the solid. Evolution laws for the damage variables (scalar or tensorial) are obtained from phenomenological or micromechanical observations. Extensions of mesloscale investigations [3,4] also led to continuum damage theories. The researchers in Refs [5-lo] and others modified these approaches to different problems. Continuum damage theories allow the easy incorporation of damage into existing continuum models. Nevertheless, it remains necessary to recognize the limits of the various continuum models used to describe damaged material behaviour. The validity range of continuum damage mechanics terminates if neighbouring voids start to coalescence, forming the first macro cracks. From thereon, fracture mechanic concepts should substitute the continuum damage model. This paper investigates the damage evolution for ductile, polycrystalline materials under quasi-static

loading. The effects of large deformations are included. Effects, primarily resulting from dynamics, such as micro-shear bands, will be neglected. The phase of micro-damage is described by the CocksAshby void-growth model. This model is coupled with a macro-crack model for the description of macro-damage. Increments and directions of macrocrack propagation will be controlled by the micro-damage evolution. In connection with experiments of ductile failure, a link between cracks and the appearance of voids was detected. This observation led to the development of a number of models which combined material damage and void-volume growth. First investigations to define a criterium for ductile failure go back to McClintock [4], when he studied the change of volume of circular and elliptical voids embedded in plastic material in the plane strain case. He described an approximate function for material failure, defining when the voids of neighbouring cells start to coalescence. Continuum damage models lose their validity after macroscale cracks arise, and then fracture mechanics models have to be used. Griffith and Irwin developed the foundations of linear elastic fracture mechanics from 1921 to 1957 [ll]. Their criteria for unstable crack propagation is based on energy balance at the crack tip. For the first time, Sneddon found approximate results for the stress distribution at the crack tip [2]. Rice and Rosengren [12] and Hutchinson [13] developed solutions for the stress field, considering plastic deformations in the crack tip region. However, their HRR-theory (HutchinsonRice-Rosengren), is restricted to small plastic strains. Only with the help of numerical procedures can large

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C. Kiinke

644

plastic deformations at the crack tip be investigated. McMeeking and Parks performed finite element computations with blunted cracks; comparing their results with the HRR-theory [14], differences in the near field predictions are discovered.

larger than the classical Cauchy stress. A necessary presumption for the effective stress concept is isotropy of the damage distribution. This means that the quotient of void area dAv and matrix material area dA, is independent of the direction of the base vectors g,:

2. MATERIAL BEHAVIOIJRIN THE MICRO-DAMAGE

dA,v dAv dAIM=a=%’

RANGE

2.1. Strain tensor Choosing a Lagrangian formulation the position vectors of point OP,Orat time ‘t and r at time t are given by Or= OXe,, r = Ye,.

dVv

(4)

Following eqn (3) we relate the effective stress tensor T to the classical stress tensor T:

(5)

(1)

Based on the velocity of the point P, which is defined as material derivative of the position vector r for fixed material coordinates Ox’,the spatial gradient L of the velocity can be written as the sum of a symmetric tensor D, called the rate of deformation tensor or stretching tensor, and a skew-symmetric tensor W, called spin tensor:

The stress tensor T is now split into a spherical tensor ~1, the hydrostatic part: p = itr(T),

and a deviatoric tensor T’, the distortional T’ = T - ~1.

(6) part, (7)

L = D + W with D = f(L + LT), w = f(L - LT).

(2)

2.2. Stress tensor Figure 1 demonstrates a deformed volume element in Lagrangian coordinates xi. We concentrate our attention to the sectional area dA, with the associated force vector df I. All other sectional areas can be seen analogously. If the volume element contains embedded voids, only the matrix material area dAm is carrying stresses. With this we define an effective Cauchy stress ?I’: df’ = r’jgdrl, = Z”g,dA,M = ?‘gi(dA, - dA,v),

(3)

To fulfill the objectivity requirements of a frameindifferent stress rate and of energy conjugacy between stresses and strains, we introduce the Jaumann stress-rate tensor: %=$-WT+TW,

(8)

with ‘I as the material derivative of the Cauchy stress tensor and W as the spin tensor of eqn (2) [15]. 2.3. Plasticity law for solids with isotropic damage distribution

The void-growth model used in this work assumes an isotropic damage description. Therefore we define microvoid area dAlv matrix area dAIM

Fig. 1. Force vector dT acting on sectional area dAl

645

Simulation of damage evolution in ductile materials

a

d

e

f Fig. 2. Macro crack propagation with the delete-and-fill method.

Di, = D, + D,,

a scalar damage variablef as the ratio between void volume and element volume:

(12)

with dVv f=dV’=

dV-dV, dV

=l--@.

dVM

(9)

Following eqns (4), (5) and (9) we obtain

-r= !/)T. (1

(10)

In classical plasticity theory the volume remains constant, strains only change the shape of an element. If we consider microvoids there might be also changes in the volume of an element dV if the microvoid volume dVv changes. Therefore the rate of deformation tensor D is split up into an elastic part D. and an inelastic part Di,: D = D, + Di,,

(11)

where the inelastic part now contains volume constant (classical plasticity) and volume changing (void-volume growth) parts:

D, = Di, - $r(Di,,)I,

D, = $r(Di&

(13)

2.3.1. Elastic deformation. Following Bruhns [16] the elastic part of the total deformation can be described as hypo-elastic material, if we assume the elastic deformations as negligible compared to the plastic ones:

De = z

1 (

v/ JL+_ tr(*)I LT + 3(1 + p)

, >

(14)

where t’ is the Jaumann rate of the deviatoric effective Cauchy stress T’, p, G is the Poisson ratio and modulus of shear, and T is the material derivation of the effective Cauchy stress T. 2.3.2. Plastic deformation (volume constant). In the, sequel the classical Levy-Mises plastic constitutive equation with isotropic hardening will be applied for the matrix material surrounding the micro-voids,

646

C. Kiinke

neglecting all influences of the embedded micro-voids on the hardening behaviour of the matrix material [8, 17, 181. Introducing the deviatoric part of the effective stress tensor T’ into the yield condition, F = ftr(T’l) - k*(wp) = 0,

spherical hole under a predefined stress state, an approximative relationship between the rate of damage f and the applied stress state can be obtained [20,21]:

(15) f=

with the hardening function k2 = fa:, depending on the plastic work W,, and applying an associative flow rule for the plastic rate of deformation,

sinh{23($))(&-(1

-l))&

(19)

where

T, = equivalent stress = ,,/%?? and plastic strain rate = &i+. Due to the observation that the closure of’vzds is more difficult than their opening, the damage process is assumed to be irreversible [9]. In the presence of compressive stresses, the void-growth rate is set to zero, assuming f as a positive monotonically increasing function. Only one material parameter n is needed to calculate the evolution equation of the internal porosity. Cocks and Ashby assign the value of the power law creep exponent to n. This value has to be determined from experimental results, e.g. n = 5.0 for aluminium and n = 4.8 for copper [IO]. Adding the single parts of the rate of deformation from eqns (14) (16) and (18) and performing some transformations, we obtain the constitutive equation for the finite element method: d,, = equivalent

DP-g

D, = 2Ll’,

(16)

1 is finally obtained as

(17)

2.3.3. Plastic deformation sidering the conservation of elastic volume deformations, between the spherical part deformation D, and the rate .A1917 D,=‘-

(volume changing). Con-

mass and neglecting the we obtain the relation of the inelastic rate of of void-volume fraction

f 3 1

-f’.

(18)

% = 2G D + 6

2.4. Void-growth model of Cocks and Ashby

The void-growth model of Cocks and Ashby is based upon micromechanical material considerations [20]. They assume that microvoids will nucleate at grain boundaries, inclusions or segregations. Spherical microvoids will grow due to inelastic deformation of the surrounding matrix material until neighbouring voids coalescence. Studying an idealized grain structure with an embedded

tr(D)I

[ -

T+pp

l+p

(1 -2p)

Ak 3(1 -f)’

>I’

A = sinh{23($)1

x

[&

- (1 -I,]$.

The Cocks-Ashby micro-damage model has been implemented into the two-dimensional finite element program FRANC-2D [22], which is a development of the Cornell University fracture group at Cornell University, New York. To verify the micro-damage model, the results obtained from this program are compared with results from related literature, based on the Gurson model [23], with satisfactory agreement. 3. MATERIAL BEHAVIOUR IN THE MACRO-DAMAGE RANGE

3.1. Coupling of the void-growth model macro-crack model

Fig. 3. FE-mesh around a blunted crack-tip.

and

a

All void-growth models lose their validity if micro-voids start to coalescence. Voids then lose their spherical shape and grow more and more penny-

647

Simulation of damage evolution in ductile materials

Load incre;mentation loop t 0 + A9 m

Load incrementation Convergence

r

iteration loop k * (t+Att)K$-1)

Tangent stiffness matrix calculation

I

Calculation of displacement increments

Calculation of the rates-of-deformation D*) out of the displacement increments

AU@)

I

AU@)

I t Calculation of stresses t=t+At t+AtT(k)

=tT

t+AtC(k-l) D&‘) dz

+ I

z=t + W(k) t+AtT(k-1)

At

_

t+AtT(k-1)

w&L)

At

Check of convergence criterium not fulfilled

fulfilled Update of displacements, stresses and damage variables

t+AtF(k)

1 Critical damage value reached?

I

Interpolation of stresses, strains, damage values on the new mesh

I

End of computation or new load step I

I

I

I

Fig. 4. Schematic overview of the computational procedure.

shaped. In general, this situation is defined as the nascent state of macro cracking [17,24], given by a value of approximately 20% void-volume fraction. At this point of time the material fails, leading to local stiffness cancellations and geometry as well as boundary condition changes. The change from micro- to macro-damage is progressive, depending on material, shape of the test specimen and loading history. Nevertheless, in the range of continuum

mechanics a discrete value for f can be used to separate the micro-damage from the macro-crack phase, similar to the treatment of plasticity, where a discrete value for the yield stress separates the plastic from the elastic range. Up to now, there exist two general approaches in the literature for simulation of micro- and macrodamage in one model. The first approach is the continuation of micro-damage theory even if the

648

C. Konke

axis of rotation po = 95 N/mm

load-step function:

load steps

lmm

Fig. 5. Axisymmetric tensile bar, system and initial FE mesh. Material = aluminium; initial void-volume fraction f= 0.05; material parameter of the Cocks-Ashby model n = 5; global loadfactor = 1.35; pa = 95 N mm-‘.

critical void-volume fraction fc is passed [18]. This approach neglects the anisotropic damage, which develops after the critical void-volume fraction has been reached. Because isotropy of the damage distribution is an essential assumption for all void-growth models, the results obtained are incorrect. In the second approach, the smeared crack approach, the stiffnesses of elements are reduced to simulate complete material failure and hence macro

cracking. This method, which is used in combination with micro-damage models for example by Tvergaard and Needleman [25] and Feldmtiller [ 171, was introduced by Rashid [26]. He studied the fracture behaviour of prestressed concrete pressure vessels. The crack is not represented explicitly, but is modelled as a “smeared crack” by modifying the material constitutive relations for the element. For concrete, a brittle material with a coarse grain

0.2009

0.2097 0.2006 0.2004

Fig. 6. Void-volume fraction f [ -1, before introducing a macro crack.

Simulation of damage evolution in ductile materials

\

Fig. 7. First crack

structure, the assumption of simulating macro-crack propagation by a number of small cracks appeals to one’s intuition of blow cracks propagate in a brittle material. In case of metals and glasses with a relatively finer grain structure, this assumption is less obvious. A further disadvantage of this method is the deterioration of the stiffness matrix condition during the solution proce:dure. Three more fundamental difficulties of the smeared crack approach were given by Wawrzynek and Ingraffea [22]:

increment.

(1) The crack geometry is described ambiguously, because there are no free crack surfaces. Without an explicit crack opening profile, analyses of some types of fracture problems-for example hydrofracturing for oil well simulations-are not possible. (2) Results for crack propagation are highly sensitive to the finite element mesh applied. If elements in the region of the expected crack path become too small, there might be a lack of convergence. On the other side a relatively fine mesh

175.9

Fig. 8. Equivalent

stress [N mm-*],

after introducing

the first crack

increment.

650

C. Kijnke

94.91 85.71 76.51 67.31 58.12

30.52 21.32 12.12 2.924

Fig. 9. Fifth crack increment, maximum shear stress [N mm-l]

has to be used in the region of the expected macro crack trajectory. (3) In case of element edges not parallel to the direction of crack propagation the phenomenon of “stress-locking” might occur, because elements in the region of the crack exhibit large artificial stresses. The overall displacements in the model might be accurate,

but stresses and strain-energies calculated within finite elements will still be inexact in the crack region. 3.2. Crack propagation delete-and-fill strategy This include

by the author’s modiJied

paper will demonstrate a new concept to micro- and macro-damage effects in one

Fig. 10. Seventh crack increment.

Simulation of damage evolution in ductile materials model [27]. After passing the limit value of the void-volume fraction fc, a macro crack is established in the model. Crack propagation is persecuted incrementally, letting the crack grow in finite crack increments. The crack propagation is controlled by the distribution of the micro-damage variablefin the crack-tip region. ,4daptively and automatically, for every crack increment well-suited new meshes are generated. All history dependent data-displacements in nodes, stresses, equivalent plastic strains and damage variables in Gauss points-is interpolated over the old mesh and transformed to the new one. In the early 1980s Saouma and Ingraffea introduced a simulation method for discrete crack propagation without regarding the existing mesh [22]. Their element delete-and-fill strategy, adapted for the coupling between a micro-damage and a macro-crack model, is demonstrated in Fig. 2, as follows. The scalar damage parameter f is calculated in every element at the Gauss points. If the damage valuefin one Gauss point reaches the critical valuef,, a macro crack is initiated from the nearest element corner node-the node with the maximum value for J A circular shaped region is then generated with its central point in the crack initiation point-the element corner node. The radius of the element cluster depends on the characteristic length of the element containing the new crack-tip. An element is deleted if one of its nodes falls inside the circular region (Fig. 2a, 1,). With the next step the new crack-tip is positioned in that Gauss point, where the critical damage value was reached (Fig. 2c) and elements are placed around the new crack-tip (Fig. 2d). In the final step, the area left between crack-tip elements and the original mesh is suitable remeshed (Fig. 2e). The last diagram shows the new open crack, demonstrating that not only the meshing, but also the geometry and the boundary conditions of the system had to be changed (Fig. 2f). With this procedure the crack always starts from an element corner node located on a boundary edge of the model. A similar concept can be used to initialize an embedded crack. The propagation of an existing crack is driven by the distribution of the damage parameter around the crack-tip. The crack will propagate if the critical damage value is reached in a Gauss point of an element adjacent to the crack-tip. The orientation of the new crack increment is determined by the maximum damage value found in an adjacent element. In case of an elastic-plastic crack propagation all results are depending on the load history. Therefore the results have to be interpolated from the old mesh onto the new one in case of a remeshing. The program coded in this work, only contains isoparametric elements with quadratic shape functions, eight-noded quadrilateral and six-noded triangular elements. Interpolation of the displacement field is based on quadratic shape functions. Bilinear shape functions are used to interpolate stresses, strains and

651

the damage variable f. These values serve as initial values for the new mesh and the next load step. All problems studied in this work show large plastic deformations at the time the macro crack is introduced into the model. For that reason we have to introduce the macro crack as a blunted crack. The finite crack opening size & depends on the crack increment size. The eight-noded isoparametric quadrilateral elements usually used are taken as crack-tip elements. One side of the elements is shrunken so that nodes one, seven and eight are positioned on the ideal semicircular blunted cracktip. Figure 3 is showing a typical mesh around a crack-tip for this case. A schematic overview over the whole computational procedure is given in Fig. 4. 4. EXAMPLE

In this section the new concept is demonstrated by an example, using the following material parameters for aluminium: E = 60000 N mm-2, 6, = 93.5 N mme2, T, = 6, + 117.2 ci586N mm-2. The critical void-volume fraction is chosen to fc = 0.20. 4.1. Axisymmetric

tensile bar

Figure 5 illustrates the axisymmetric tensile bar system under axial loading conditions and the initial finite element mesh. In this example a coarse initial mesh is chosen to demonstrate that a quite good prediction of the macro crack path can be found with a very “bad” initial mesh. Further investigations on how the results for the history dependent variables are influenced by the mesh density have to be done. The radius of the system is reduced by 1% in a certain height, to enforce the initiation of the macro crack in this height. This leads to a concentration of stresses in this plane and thus to an inhomogeneous damage distribution. The same effect could be achieved by inhomogeneities in the initial void-volume distribution, e.g. by a small increase of the initial damage value in a certain region. This procedure is reasonable with respect to real inhomogeneous initial void-volume distributions. The actual loading in every step is calculated by multiplying the load p0 with the factor of the load-step function and the global load factor. In Fig. 6, the concentration for the void volume fraction in the plane of radius reduction by 1% can be seen. Figure 7 demonstrates the onset of the first macro crack increment in a sequence with the mesh adaption. Comparing Figs 6 and 7 we observe that the macro crack initiates from the point of the maximum void-volume fraction f. Then the crack grows from the axis of symmetry to the outer which agrees with the results from boundary, Refs [17] and [25]. Figure 8 depicts the equivalent. stress distribution at the end of the next load step demonstrating a clear stress reduction on the lower left side of the macrocrack. The crack path develops

652

C. Kiinke

in a zigzag fashion, in planes which are inclined by approximately 45” against the horizontal line. While the direction of the first crack increment is not only influenced by the micro-damage distribution, but also by the procedure described above, the directions of all following crack increments are completely independent of the actual FE meshes. The zigzag crack trajectory is confirmed by experimental results [28]. Shortly before the crack reaches the outer boundary, the final shear failure of the residual cross-sectional area develops. This shear failure takes place in the maximum shear stress plane. The maximum shear stress distribution, after introducing the fifth crack increment, is shown in Fig. 9. “Shear bands” inclined by approximately 45” against the horizontal line are visible. Figure 10 demonstrates the change in geometry and mesh for the seventh crack increment. The crack path leaves its zigzag path and tends in the direction of the maximum shear stress.

the initiation state of a macro crack should be defined, as close as possible to physical reality. REFERENCES

1. J. Hult, Introduction and General Overview, Continuum Damage Mechanics, Theory and Application (Edited by D. Krajcinovic and J. Lemaitre), pp. l-36. Springer, Berlin (1987). 2. U. Graf, Bruchmechanische Kennwerte und Verfahren fiir die Ermiidungsfestigkeit GeschweiBter Aluminiumbauteile. Berichte aus dem Konstruktiven Ingenieurbau Technische Universitat Miinchen, Miinchen (1992). 3. A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. engng Mater. Technol. 99, 2-15 (1977).

4. F. A. McClintock, A criterion for ductile fracture by the growth of holes. J. appl. Mech. 35, 363-371 (1968).

5. V. Tvergaard, A. Needleman and K. K. Lo, Flow localization in the plane strain tensile test. J. Mech. Phys. Solids 29, 115-142 (1981).

6. V. Tvergaard, InlIuence of void nucleation on ductile .

shear fracture at a free surface. J. Mech. Phys. Solids 30, 399425

5. SUMMARY

AND CONCLUSION

The Cocks-Ashby void-growth model for the simulation of micro-damage evolution in isotropic, polycrystalline and ductile materials-such as steel or aluminium-is implemented into the two-dimensional finite element program FRANC-2D. This model is coupled with a new model for macro-crack propagation. At the time neighbouring micro voids start to coalescence, the initiation of a macro crack starts. Therefore a macro crack is introduced into the model after the critical void-volume fraction has been reached. Macro crack propagation is controlled by the distribution of the void-volume fraction in the crack-tip region. The simulation of the macro-crack propagation is carried out by an extended element delete-and-fill algorithm, independent of the initial mesh. In an incremental and adaptive way a special mesh is generated for every new geometry. The macro crack is introduced as a blunted crack taking large plastic deformations into account at the initiation time. The physical problem of macro-crack initiation and macro-crack propagation is properly described by the introduction of a macro crack into the numerical model. Both crack faces are introduced as free surfaces with exact boundary conditions. Comparing the own stress distribution results with results from literature based on the “smeared crack” approach, it can be seen that the new concept of coupling micro-damage and macro cracks gives more realistic stress predictions [17]. The coupling model, developed in this work, enables the study of complete damage evolution from nucleation and growth of first micro-voids to the range of macro-damage. The advantage of this method lies in the independence of a chosen crack initiation geometry. Nevertheless, the question remains open for further research on how

(1982).

I. V. Tvergaard, Material failure by void coalescence in localized shear bands. Int. J. Solids Struct. 18. 659-672 (1982). 8. V. Tvergaard, Material failure by void growth to coalescence. Adu. appl. Mech. 27, 83-151 (1990). 9. K. K. Mathur and P. R. Dawson, Damage evolution modeling in bulk forming processes. In: Computational Methods for

Predicting

Material Processing

Defects

(Edited by M. Predeleanu), pp. 251-262. Elsevier, New York (1987). 10. K. K. Mathur and P. R. Dawson, On modeling damage evolution during the drawing of metals. Mech. Mater. 6, 179-186 (1987).

11. K. Heckel, Einfiihrung in die Technische Anwendung der Bruchmechanik. Carl Hanser, Miinchen (1983). 12. J. R. Rice and G. F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material. J. Mech. Phys. Solids 16, 1-12 (1968). 13. J. W. Hutchinson, Singular behaviour at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13-31 (1968). 14. T. L. Anderson, Fracture Mechanics, Fundamentals and Applications. CRC Press, Cleveland, OH (1991). 15. L. E. Malvern, introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ (1969). 16. 0. T. Bruhns, Einige bemerkungen zur beschreibung inelastischer prozesse im bereich hoher deformationsgeschwindigkeiten. Zeit. Angew. Math. Mech. 67, T181-T183 (1987). 17. A. Feldmiiller, Ein thermoplastisches stoffgesetz isotrop geschldigter kontinua. Mitteilungen aus dem Institut fiir Mechanik no. 75, Ruhr-Universitat Bochum, Bochum (1991). 18. S. T. Tjotta, Formability and the growth of damage. In: Numerical Methods in Industrial Forming Processes (Edited by J. L. Chenot, R. Wood and 0. C. Zienkiewicz), pp. 187-192. Balkema, Amsterdam (1992). 19. D. C. Leigh, Nonlinear Continuum Mechanics. McGraw-Hill, New York (1968). 20. A. C. F. Cocks and M. F. Ashby, Intergranular fracture during powerlaw creep under multiaxial stresses. Metal Sci. 395402

(1980).

21. J. B. Martin, A note on the determination of an upper bound on displacement rates for steady creep problems. J. appl. Mech. 33, 216-217 (1966). 22. P. A. Wawrzynek and A. R. Ingraffea, Discrete

Simulation

of damage

evolution

modeling of crack propagation. Theoretical aspects and implementation issues in two and three dimensions. Department of Structural Engineering, Report no. 91-f;, Cornell University, Ithaca, New York (1991). 23. B. Bennani, P. Picart and J. Oudin, A finite element algorithm for micro-void nucleation, growth and coalescence. Inr. J. Damage Mech. 2, 118-136 (1993). 24. V. Tvergaard, Papers related to IUTAM-CISM Summer School, September 1991. Modeling of defect and fracture mechanics. Notes and papers relating to the five lectures. (1991). 25. V. Tvergaard and A. Needleman, Analysis of the

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cupcone fracture in a round tensile bar. Acta Metal. 32, 157-169 (1984). 26. Y. R. Rashid, Ultimate strength analysis of prestressed concrete pressure vessels. Nucl. Engng Des. 7, 334344 (1968). 27. C. Kiinke, Coupling of a micromechanically based void-growth model and a macro-crack model for the simulation of damage evolution in ductile materials. IKIB Techn-wiss. Mitt. no. 944, Ruhr-Universitlt Bochum, Bochum (1994). 28. G. Lange, Systematische Beurteilung Technischer SchadensWlle. In: Lehrinhalt und Vortragstexte e. Fortbildungsseminars d. Dt. Ges. fur Metallkunde e.V. (1983).

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