Eindhoven University of Technology

MASTER

Higher-order plasticity formulations for ductile failure

Engelen, R.A.B.

Award date:1999

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Higher-order plasticity formulations for ductile failure

Engineering thesis committee:

Prof.dr.ir. F.P.T. Baaijens Drjr. M.G.D. Geers Dr.ir. W.A.M. Brekelmans Dr.ir. P.J.G. Schreurs Ir. R.H.J. Peerlings

February 1999

mate

Roy A.B. Engelen Master's thesis MT 99.004

(chairman) ( supervisor)

(advisor)

tea Section of Materials Technology Faculty of Mechanical Engineering Eindhoven University of Technology

Summary

For many engineering materials the failure behaviour on the macroscopic level is notably influenced by the interactions and mechanisms on the micro-structural level. Due to the miniaturization in the last few decades in many manufacturing processes, the micro-structural influence becomes more apparent and the need for alternative material models arises.

This thesis is focused on the modeling of ductile failure in materials. For classical plas­ticity formulations using a softening constitutive relation, the governing differential equation of the boundary value problem turns ill-posed at the onset of softening. Within a finite ele­ment framework this leads to unphysical solutions and the process of ductile failure cannot be simulated properly.

In literature many formulations are proposed in order to restore the well-posedness of the boundary value problem. For this purpose most formulations adopt a length parameter, which can be related to the micro-structural mechanisms and the associated process zone within a material. The most efficient among these methods, the gradient-enhanced models, generally incorporate the Laplacian of a local quantity in the constitutive equation.

First, several gradient-enhanced plasticity models from the literature are treated and their numerical difficulties are elucidated. To cope with these problems an alternative formulation is proposed, which adopts a nonlocal quantity in the yield function. The relation between the nonlocal and the local quantity is governed by a differential equation of the Helmholtz type, which is solved in a coupled fashion in addition to the equilibrium condition. However, one­dimensional simulations reveal unavoidable and unphysical oscillations of the local quantity.

Based on the introduction of a ductile damage parameter, a second formulation is pro­posed. In this formulation the yield stress depends on the product of both a function of a local and a function of a nonlocal quantity. A one-dimensional finite element discretization is derived, and the efficiency of this formulation to describe ductile failure is investigated. Finally, a brief discussion is made on the incorporation of different types of history dependent behaviour of the ductile damage parameter.

III

Samenvatting

Voor vele constructie materialen wordt het schade gedrag op een macroscopische schaal merk­baar be'invloed door interacties en mechanismen op het niveau van de microstructuur. Ten gevolge van de miniaturisering gedurende de laatste jaren van de fabricage processen, wordt deze invloed meer en meer duidelijk en ontstaat de behoefte aan nieuwe materiaalmodellen.

Dit verslag richt zich op het modelleren van het ductiele schadegedrag in een materiaal. Bij klassieke plasticiteitsmodellen, die gebruik maken van een softenend gedrag in de consti­tutieve relatie, raakt de beschrijvende differentiaalvergelijking van het randwaarde probleem slecht geconditioneerd bij het optreden van softeningsgedrag. Dit leidt tot incorrecte resul­taten bij het simuleren van het process met een eindige elementen methode.

In de literatuur worden vele methoden besproken om de goede conditionering van het randwaarde probleem te herstellen. Hiervoor maken de meeste modellen gebruik van een lengte parameter, die kan worden gerelateerd aan de mechanismen en het procesgebied op het microstructurele niveau van het materiaal. De meest efficiente onder deze methodes zijn de gradient afhankelijke plasticiteits modellen, die in het algemeen de Laplaciaan van een lokale grootheid in de constitutieve omschrijving betrekken.

Allereerst worden enkele gradient afbankelijke modellen uit de literatuur onderzocht en worden de voornaamste numerieke problemen ervan toegelicht. Om deze problemen te voor­komen wordt een alternatief model geintroduceerd met een niet-Iokale grootheid in haar vloei­functie. De relatie tussen de lokale en de niet-lokale grootheid wordt beschreven aan de hand van een differentiaal vergelijking van het Helmholtz type, die op een gekoppelde manier met het krachtenevenwicht wordt opgelost. De resultaten van een-dimensionale simulatie vertonen echter onvermijdbare en niet-fysische oscillaties van de lokale grootheid.

Gebaseerd op de invoering van een ductiele schade parameter wordt een tweede model voorgesteld. In deze formulering hangt de vloeispanning af van het produkt van zowel een functie van een lokale als van een niet-Iokale grootheid. Een een-dimensionale discretisatie wordt afgeleid en de efficientie van de formulering onderzocht om ductiel schade gedrag te modelleren. Ter afsluiting wordt een korte discussie gehouden over het implementeren van verschillende types geschiedenisafbankelijk gedrag van de ductiele schade parameter.

v

Contents

Summary

Samenvatting

Notations

1 Introduction 1.1 Strain localization and ductile failure. 1.2 Scope and objectives of the research . 1.3 Assumptions and outline of the thesis

2 Analysis of gradient-enhanced plasticity formulations 2.1 Local plasticity .............................. . 2.2 Gradient-Enhanced plaSticity ...................... .

2.2.1 Yield function dependent on the Laplacian of a local quantity. 2.2.2 Yield function with nonlocal yield strength . . . . . . . . 2.2.3 Yield stress dependent on the nonlocal plastic multiplier.

3 Gradient-enhanced plasticity based on ductile damage 3.1 Yield condition and stabilization ..... 3.2 Rate constitutive equations . . . . . . . . 3.3 Weak formulation of differential equations 3.4 Finite element implementation ..... .

3.4.1 Discretization of the weak form of the governing equations. 3.4.2 Consistent tangent stiffness matrix and return-mapping

4 Numerical results 4.1 Ductile damage evolution law

4.1.1 Linear evolution .. . 4.1.2 Exponential evolution

4.2 Mesh-independency...... 4.3 Analysis of localization behaviour . 4.4 Cyclic deformation process 4.5 Discussion ............. .

5 Conclusions and recommendations

References

VII

iii

v

ix

1 1 3 4

5 5 7 8 9

11

15 15 16 18 19 19 21

25 25 25 27 27 30 32 34

35

31

Notations

Throughout the theoretical elaborations the following notations will be used, with the related definition in a Cartesian coordinate system between parentheses:

Vectors and tensors

Quantities

Operators

Scalar Vector Second-order tensor Fourth-order tensor

Dyadic product Inner product Double inner product Gradient operator nth Order gradient Divergence operator Laplacian operator Conjugation Inversion Finite difference Infinitesimal variation Time derivative

lX

a

AB A·B A:B V vn V· \72

AC

A-I

LlA 8A A

ai) (= Aij)

Aijkl)

(= AijBkl) AijBjk)

(= AijBji)

( A~j = Aji )

x

Matrices and columns

Quantities

Operators

Scalar Column Matrix

Matrix product A B Transposition AT Inversion A-1

Finite difference LlA Infinitesimal variation 8A Time derivative A

Notations

Any notation which has not been explicitly defined in the previous paragraphs, will be ex­plained at the first point of use in the remaining part of this thesis.

Chapter 1

Introduction

1.1 Strain localization and ductile failure

Experimental observation of the plastic deformation of various engineering materials reveals the existence of localization phenomena. Strain localization is a notion describing a de­formation mode, in which the whole deformation concentrates in a narrowing process zone (e.g. one or more narrow bands), while the rest of the structure usually unloads. The extent of localization depends on material properties, geometry and boundary conditions.

Ductile failure is the notion of gradual structural softening, which is confined to a certain localization zone. On micro-structural material level this can be seen as the development and growth of micro-voids into a certain zone until the material fails gradually. Under plas­tic loading conditions lattice imperfections will travel through the material (e.g. metals). After a certain amount of hardening due to the geometrical obstruction of these dislocations, locally micro-voids arise. Micro-voids decrease the local material strength and facilitate sub­sequent (plastic) deformation. Consequently, micro-voids tend to concentrate in a certain region. Other phenomena which give rise to failure at micro-structural level are inter-granular or intra-granular deformations, deformations of polycrystals, micro-structural instabilities (e.g. phase changes), inter-granular cleavage etc. for metals. And for polymers impurities or the breakage of bonds between molecule chains.

For classical plasticity within a continuum framework, local failure phenomena (e.g. soft­ening) is included by adopting a decreasing yield strength in the yield condition. However, the mathematical implication of using such a constitutive relation is the loss of material sta­bility coupled to an ill-posed initial or boundary value problem, and the governing differential equation locally changes type (e. g. elliptic to hyperbolic for quasi-static problems). The loss of well-posedness triggers the solution with the smallest energy dissipation, and all plastic deformations will tend to concentrate in the smallest volume, which is one element for a discretized continuum. In consequence, the solution of numerical simulations fully depends on the direction and the fineness of the finite element discretization, leading to meaningless results.

1

2 Chapter 1 Introduction

In order to overcome these deficiencies and restore the well-posedness of the boundary value problem, a number of approaches have emerged in the past decades:

• fracture energy methods (PIETRUSZCAK & MROZ [17]) relate material parameters to the element size such that a higher specific energy dissipation occurs in smaller elements; this method displays the correct global response without actually restoring the well­posedness of the boundary value problem;

• viscous regularization methods preserve the well-posedness through the use of a fluidity or viscosity parameter in the constitutive model. This type of regularization is both effective in static analysis (no inertia effects) and dynamic analysis. However, this type of regularization cannot be used for rate-independent materials or loading conditions (SLUYS & DE BORST [19]; GEERS et al. [7]; WANG & HABRAKEN [24]); Furthermore, it has been shown by WANG [23] that the solution is more sensitive to imperfections than the solution obtained from nonlocal or gradient models;

• the Cosserat micro-polar continuum theory uses additional, rotational degrees of free­dom and a couple-stress to describe the forces exerted by one material element on another (STEINMANN & WILLAM [20]; DE BORST & MUHLHAUS [1]).

• nonlocal (integral) continuum models are based on nonlocal variables which are spatially averaged variables of local quantities in a volume in the vicinity of a material point (DE VREE et al. [5]; STROMBERG & RISTINMAA [21]; POLIZZOTTO et al. [18]);

• gradient-enhanced models explicitly or implicitly include higher-order deformation gra­dients in the constitutive model (MUHLHAUS & AIFANTIS [13]; PEERLINGS et al. [16]; XIKUI & CESCOTTO [26]; MIKKELSEN [12]; SVEDBERG & RUNESSON [22]; NIX & GAO [14]; DE BORST et al. [4]).

With the exception of the fracture energy approach, these formulations incorporate an intrinsic length scale. This length scale is a micro-structural dependent material parameter that is used in the continuum description. It is related to the micro-structure and the in­teracting of the flow or failure events on that level. From the mathematical point of view the length scale restores the well-posedness of the boundary value problem. This regularizing effect has been illustrated for a gradient-dependent yield function using a one dimensional analytical analysis [2].

Among the regularizing techniques mentioned above, the (integral) nonlocal and gradient­enhanced formulations are most efficient. Various models of this type are still subject of outstanding research. Nonlocal and gradient models are strongly related, but the gradient­enhanced models use a differential description of the governing equations which is more attractive within a finite element formulation. In order to avoid misunderstanding a gradient­enhanced formulation is considered to be implicit regarding to the degrees of freedom of the discretized set of governing equations. No distinction is made whether these quantities are local or nonlocal.

The numerical treatment of explicit gradient-enhanced formulations generally requires a numerically expensive CI-continuous interpolation and consequently restricts the adopted element type severely. However, a CO-continuous interpolation suffices when an extra set of

Chapter 1 Introduction 3

degrees of freedom is discretized (PAMIN [15]; DE BORST & PAMIN [3]). The Laplacian of the plastic multiplier is used in the yield function. In addition the gradient of the plastic multi­plier is discretized besides the displacements and plastic multiplier. However, this algorithm shows some numerical difficulties, which result in a number of additional requirements with respect to the discretization.

An implicit gradient-enhanced model has been proposed by MEFTAH et al. [11] which requires a CO-continuous interpolation using only two sets of degrees of freedom. The yield function depends on the nonlocal yield stress, which acts as the source term in the differential Helmholtz equation. Apart from the reduction of degrees of freedom, the problem of handling the elasto-plastic boundary remains. Furthermore, the local yield stress reaches the zero-level prematurely, e.g. before final failure.

1.2 Scope and objectives of the research

This thesis focuses on gradient-enhanced plasticity models and will discuss both the advan­tages and disadvantages of a number of existing formulations with the intention to propose a new variant of a gradient-enhanced plasticity modeL For this purpose an implicit formulation will be used, where the yield function is enriched with a nonlocal quantity. An additional differential equation of the Helmholtz type is solved besides the equilibrium condition in a coupled fashion.

The main objective is to construct a numerically efficient algorithm for the analysis of plastic behaviour and ductile failure in softening materials within a finite elements context. A primary requirement is the regularization of the strain localization to prevent the loss of well-posedness and its negative consequences in numerical analysis, e.g. pathological mesh dependence. But another major concern is the numerical efficiency, as the algorithm has to be well-suited for three-dimensional analysis, large deformations and coupling with elasticity­based damage in future work.

A first formulation incorporates a yield stress which additively depends on a function of the nonlocal plastic multiplier. The differential Helmholtz equation is formulated in an im­plicit manner with respect to the nonlocal quantity, similar to the approach that has been adopted in some gradient-damage models (GEERS [6]; PEERLINGS et al. [16]). However, nu­merical solutions display unphysical oscillations in the plastic zone. Using a one-dimensional analysis these oscillations can be explained.

An alternative formulation incorporates a ductile damage variable that progressively re­duces the yield stress. The ductile damage variable is a function of the nonlocal plastic multiplier and provokes softening, while the local plastic multiplier introduces hardening. This formulation is more flexible and the unphysical oscillations vanish.

4 Chapter 1 Introduction

1.3 Assumptions and outline of the thesis

In this work quasi-static loading and small deformations are assumed. Further, a length scale which is spatially isotropic is used, so the interactions on micro-structural level are consid­ered to be independent of the direction. For one-dimensional finite element simulations this restriction is less severe than for three-dimensional problems. For the sake of simplicity only associated plastic flow is considered.

Chapter 2 analysis several gradient-dependent plasticity formulations, which are proposed in literature. The numerical difficulties which appear are exhibited. Finally, a new plasticity formulation is proposed and the problems that were encountered during simulations are elu­cidated.

Chapter 3 proposes a second plasticity formulation, which is based on ductile damage. First, the 'idea' behind the formulation is mentioned. Second, the weak formulation of the coupled set of differential equations is derived. Next, this weak formulation is discretized for a one-dimensional problem within a finite element framework. Aspects as a consistent stiffness matrix and the return-mapping of the stress are also considered.

Chapter 4 shows numerical results of the formulation derived in chapter 3. Both the in­fluence of the model parameters and the capability to simulate ductile failure are discussed.

In Chapter 5 the conclusions are summarized together with several remarks concerning future work.

Chapter 2

Analysis of gradient-enhanced plasticity formulatiolls

The classical theory of local plasticity in a continuum description serves as a starting point for the theoretical outline of the gradient-enhanced plasticity formulations. First, some basic equations of local plasticity are recalled in section 2.1 to emphasize the used symbols and the applied conventions. An analysis of some gradient-enhanced plasticity formulations proposed in literature are elucidated in section 2.2.

2.1 Local plasticity

In classical plasticity, it is assumed that the total deformation consists of both an elastic and an inelastic (plastic) fraction:

(2.1)

with e the infinitesimal strain tensor:

1 [_ {_ }C] _ e = 2" 'Vii + 'Vii = 'V8ii (2.2)

where ii denotes the displacement vector of a material point and VS is the symmetric gradient operator. For simplicity only isotropic material behaviour is considered, so Hooke's Law can be used for the elastic response:

(2.3)

where (j denotes the Cauchy stress tensor and 4C the fourth-order Hookean stiffness tensor:

-=--_::-vE-::--_-,- [II + 1- 2v 41] {I + 1I} {I 211} 1I

(2.4)

in which E denotes Young's modulus, 1I designates Poisson's ratio and 1 and 41 represent the second-order identity tensor and the fourth-order symmetric identity tensors respectively.

5

6 Chapter 2 Analysis oj gradient-enhanced plasticity formulations

In a plastic state (eP =1= 0) the instantaneous stress situation is inferred from the satisfac­tion of the yield condition:

(2.5)

where i1 is an equivalent stress (e.g. (J' - V!ud : ud for von Mises plasticity with

u d = u -1 [u: I] I as the deviatoric part of the stress) and C1y the yield stress. The ef­fective plastic strain cp, which is a nonnegative scalar function of the amount of plastic flow, is used as an internal history dependent hardening variable. The definition of the effective plastic strain reads:

t

C:p(t) = f Ep dr 'h' ~'P .• p WIt cop _t!:' t!:' '" 3"" ... 7"=0

The evolution of the plastic strain is governed by the associated flow rule:

eP = 'tN (u, C:p )

(2.6)

(2.7)

with "'I the plastic multiplier (or consistency parameter) denoting the magnitude of plastic flow and N U the associated direction. For von Mises plasticity, yield function (2.5) and monotonic loading the plastic multiplier "'I is equal to the effective plastic strain C:p.

During plastic flow (eP f= 0) the consistency condition applies:

j (iT, 't) = N : iT - h't = 0 with h = ~(J'y = hardening modulus UEp

(2.8)

In order to distinguish loading or unloading the Kuhn-Tucker conditions apply:

f '5: 0, 'tf 0 (2.9)

During computations the incremental stress values are determined using a classical return­mapping algorithm:

with u ep = 4C : eCH1 ) (2.10)

where the supscript (i+ 1) denotes the increment number . ..1"'1(H1) is determined by projecting the elastic prediction of the stress u ep back to the yield surface (I(H1) = 0).

A major drawback of local plasticity models is the loss of well-posedness when exhibiting softening behaviour. Under the assumption of small deformations, this can only occur if the material stability has been lost (WILLAM & ETSE [25]). A material is defined stable if its constitutive relationship fulfills the condition of a nonnegative second order work density (HILL [10])

82W = !8u: Oe > 0 2 - {2.11}

A sufficient condition for the structural stability of body n reads:

~ In Oe : OU dn ;::: 0 (for all kinematically admissible oc:) (2.12)

Chapter 2 Analysis of gradient-enhanced plasticity formulations 7

The loss of material stability in case of ductile failure for the classical local plasticity theory follows from the one-dimensional consistency condition. For a monotonic loading situation this can be illustrated using the second order work corresponding to the plastic strain, as only the inelastic part of the deformation works destabilizing:

-+ 62Wp = ~h f~ [&P]2 dx < 0 if h < 0(2.13) 2

where the body occupies the volume :::; x :::; ~ and the plastic deformation is confined to -~ :::; x :::; ~. This loss of stability may lead to the loss of well-posed ness of the boundary value problem.

2.2 Gradient-Enhanced plasticity

Gradient-enhanced plasticity models in general introduce a gradient term to the yield function:

(2.14)

where g is a parameter which determines the gradient influence. Motivated by a one­dimensional analytical solution for h < 0 and a constant influence [15J, the parameter is chosen as follows:

(2.15)

where £ is an internal length parameter. The consistency condition {2.8} then transforms to:

(2.16)

In order to illustrate the mathematical influence of the gradient term, again the second order work associated with the plastic strain for a one-dimensional tension bar can be evaluated:

L

62Wp = ~ f: 6f:P 6(1 dx = 2 (2.17)

where the boundary condition 6f:P = 0 for x = has been used. For h < 0 we observe that the second (gradient) term of (2.17) acts as a stabilizer which restores material stability and the well-posedness of the boundary value problem.

Within this framework a number of algorithms can be constructed. In order to get an im­pression of the regularizing effect and the numerical problems which arise, a one-dimensional tension bar is used throughout all the examples. The configuration of this bar is depicted in figure 2.1.

8 Chapter 2 Analysis of gradient-enhanced plasticity formulations

u

x L

Figure 2.1: left: Schematic representation of the one-dimensional example; the displacement u is imposed; L is the total length of the bar and d is the length of the weakened zone, where the initial yield strength is reduced with 10%; right: local linear softening behaviour

2.2.1 Yield function dependent on the Laplacian of a local quantity

The gradient-dependency of equation (2.14) was elaborated by PAMIN [15] and DE BORST & PAMIN [3]. With respect to the local yield stress a y a linear softening be­haviour is adopted for simplicity as depicted in the right part of figure 2.1. The slope is constant, denoting the softening modulus h = ~ = constant. Within the yield function the gradient dependent yield stress is defined as:

(2.18)

Besides the equilibrium equation, a second differential equation has to be solved, i.e. the consistency condition. The solution of the latter equation within a finite element context, inevitably necessitates the discretization of the plastic multiplier. Thus, deriving a weak formulation of the set of differential equations, the integral of the consistency condition is applied to the plastic part of the deformed body only. However, in order to facilitate the discretization in a standard finite element framework, the equilibrium equation and the con­sistency condition are solved with respect to the same volume. This can be done using the Kuhn-Tucker conditions:

l' ~ 0, f ~ 0, l' f = 0 (2.19)

in combination with an additional boundary condition for the plastic multiplier on the elasto­plastic boundary of the Neumann or Dirichlet type.

To satisfy the Neumann boundary condition, Hermitian shape functions (CI-continuous) are necessary. This type of discretization restricts the admissible element topology and is less flexible for the geometrical modeling of real applications. The boundary condition vanishes by adopting the gradient of the plastic multiplier tp as an (additional) degree of freedom besides the displacements u and the plastic multiplier /. In order to limit the number of equations describing the boundary value problem, the additional constraint tp "IT is enforced using a penalty approach. In this manner an attractive CO-continuous formulation was obtained. CO -continuous shape functions are continuous across the element boundaries, but their spatial first derivatives may show a finite discontinuity.

The effect of the gradient term can be elucidated using the mentioned one-dimensional example in figure 2.1. A typical numerical result for the one-dimensional bar with a weakened zone is depicted in figure 2.2. The algorithm stabilizes the localization behaviour by reducing

Chapter 2 Analysis of gradient-enhanced plasticity formulations 9

1 - __ I

I

I II I

u o L x

Figure 2.2: Stabilizing effect of an additional gradient term; in areas I the elastic elements are weakened since ug < uy; in area II the plastic elements have a higher yield strength as ug > uy

the deformation in area II with respect to the local yield stress and extending the plastic zone by weakening the elastic elements in areas I. A finite volume is thus involved in the ductile failure process, which leads to a non-zero energy dissipation. This allows the boundary value problem to remain well-posed during calculations.

A disadvantage of this algorithm is the need for small load increments. This is particularly the case when the gradient dependent yield stress reaches zero. In the one-dimensional case, the Laplacian of the plastic multiplier is positive near the plastic region, which leads to a negative gradient contribution to the yield stress:

V2ry > O} (~) h < 0 CIg < CIy (2.20)

as can be seen for elements in area I near the plastic zone II in figure 2.2. During the iterative procedure negative values of CI g arise when CI g -+ O. This may cause numerical problems when the elasto-plastic boundary is positioned within one element, e.g. the value of the yield function in the integration points has opposite signs. Because of the weak formulation, the negative values are concealed due to the averaged contribution to the right-hand side of the boundary value problem. This leads to an improper correction of the local plastic multiplier during the return-mapping of the stress, which disturbs convergence. However, this can be avoided adopting additional restrictions to the shape functions used for the element discretization.

2.2.2 Yield function with nonlocal yield strength

In contrast with the previous formulation, this method introduces a nonlocal yield stress into the yield function, as proposed by MEFTAH et. al [11]:

(2.21)

where a nonlocal variable is denoted with a bar above the symbol of the local variable. In order to obtain a relation for the nonlocal yield stress, the definition of a nonlocal quantity

10 Chapter 2 Analysis of gradient-enhanced plasticity formulations

from the integral nonlocal plasticity formulations [21] is used. A nonlocal field, a, in a body o is considered as a weighted average of the local field in the surrounding volume Os of a considered material point x:

(2.22)

where g is a weighting function and (the relative position vector within the body with re­spect to the material point x. The applied weighting function, which incorporates a length scale £, and weighting volume Os determine the micro-structural mechanical influences and interactions of the neighbouring points on the considered material point.

In order to facilitate a finite element discretization, the integral equation has to be trans­formed into a differential formulation, conform gradient-enhanced damage methods [16]. Expanding the local yield strength u y into a Taylor series, applying an isotropic weight­ing function, computing the integrals and assuming isotropic material behaviour, equation (2.22) in terms of the yield strength can be written as:

for n = 2, 3, (2.23)

where the length parameter £ depends on the selected weighting function. Neglecting the higher-order terms with n ;::: 2, yields the following relation:

(2.24)

Considering the local yield strength as the unknown quantity, this equation is an implicit differential equation of the Helmholtz type, in which the nonlocal yield strength jjy acts as a source term. The consistency condition is used to determine the value of the nonlocal quantity in relation to the equilibrium. MEFTAH et. al [11] solved the averaging Helmholtz equation in combination with the equilibrium condition, and obtained a CO-continuous mixed formulation with nodal displacements u and local yield stresses u y as degrees of freedom.

A typical numerical result of the one-dimensional bar with a weakened zone (figure 2.1) is shown in figure 2.3. Despite the advantage of a reduced number of degrees of freedom similar numerical problems arise. In the plastic zone the local yield stress turns negative prior to the nonlocal yield stress, as can be seen in figure 2.3. This leads to an ill-conditioned set of equations using the assumption

(2.25)

where h denotes the negative softening modulus, which is only valid during the softening flow (uy > 0).

Also the position of the elasto-plastic boundary within an element and the expansion of the plastic zone, disturbs the convergence of the iterative process. This due to the presence of the value of the yield function in the right-hand side of the boundary value problem, so only small load increments are admissible, similar to the previous formulation.

Chapter 2 Analysis of gradient-enhanced plasticity formulations 11

- --I

I

u o L x

Figure 2.3: Stabilizing effect of nonlocal yield stress; in areas I the elastic elements are weakened as O'y < O"y; in area II the plastic elements have a higher yield strength as O'y > O"y

2.2.3 Yield stress dependent on the nonlocal plastic multiplier

A first attempt to avoid the numerical difficulties of the two previous methods, is the use of the following gradient-enhanced yield function:

(2.26)

The differential relation between the local and nonlocal quantity {equation {2.24}} is modified in order to obtain an implicit relation for the nonlocal plastic multiplier, as suggested by PEERLINGS et al. [16] in a gradient-enhanced damage context:

(2.27)

which has to be solved in combination with the equilibrium condition. In contrast with equation (2.24), the nonlocal quantity is the variable and the local quantity acts as the source term. The return-mapping of the stress to the yield surface is used to determine the value of the local source term in relation to the equilibrium .. The opposite sign in front of the Laplacian is due to the strain based formulation in contrast with the formulation based on the yield stress. The solution of (2.27) requires additional boundary conditions at the boundary r. Most applications based on a gradient formulation for plasticity (as well as for quasi-brittle damage), use the natural Neumann boundary condition as proposed by MUHLHAUS AND AIFANTIS [13]:

(2.28)

where it denotes the unit outward normal to the boundary. Within a thermodynamically consistent framework SVEDBERG AND RUNES SON [22] justified this boundary condition for ductile materials.

When discretizing the averaging equation besides the equilibrium condition, a CO-continuous finite element formulation is obtained with the nonlocal plastic multipliers ;y as secondary degrees of freedom in addition to the displacements u.

12 Chapter 2 Analysis of gradient-enhanced plasticity formulations

In order to illustrate the implications of the proposed algorithm, a typical solution for the one-dimensional bar with a weakened zone is shown in figure 2.4. In this figure, also the evolution of the yield strength, which depends on the local plastic multiplier, is depicted.

0'

I IT I

u o L x

Figure 2.4: Stabilizing effect of nonlocal plastic multiplier; in areas I the elastic elements are weakened as <711 en < <711 (-r); in area II the plastic elements have a higher yield strength as <711 ('Y) > <7y (-r)

Since the yield stress, which is incorporated in this formulation, solely depends on the non­local plastic multiplier, the instantaneous stress can turn zero without causing numerical problems. This becomes clear when comparing figure 2.3 with figure 2.4 and noting that for this algorithm:

(2.29)

The above equation remains valid until the stress turns zero, in contrast with equation (2.25) which is violated before the stress turns zero.

Unfortunately another disadvantage appears. The consistency condition, which is applied in the plastic region, restricts the evolution of the nonlocal plastic multiplier in this region in such manner that it is hard to meet with the diffuse solution of a Helmholtz equation (2.27). Especially for the one-dimensional situation this becomes apparent for a weakened tensile bar, monotonic loading and a constant softening modulus. In the plastic zone, the consistency condition for the bar reads:

1=0 (2.26)

-+ 0& = h oiy ax ax (2.30)

Due to the equilibrium equation the following condition is applied to the stress throughout the bar:

00' = 0 ax

0& = 0 ax (2.31)

Chapter 2 Analysis of gradient-enhanced plasticity formulations 13

Evaluating the last two equations shows that the yield condition forces a constant value of the nonlocal plastic multiplier in the plastic zone. Figure 2.4 indicates the variation of the nonlocal plastic multiplier as the expected solution of the averaging equation (2.27) in the plastic zone II, which is not in accordance with the yield condition (1' is not constant in II). Within a finite element framework this inevitably induces oscillations in the local plastic mul­tiplier field, which leads to unphysical solutions.

In the next chapter a more flexible implicit gradient-enhanced algorithm is proposed which is more convenient to use and which leads to realistic strain profiles.

Chapter 3

Gradient-enhanced plasticity based on ductile damage

In this chapter an alternative gradient-enhanced plasticity formulation is introduced, which encompasses most problems cited in the previous chapter. Both the rate formulation and the implementation in a finite element framework will be treated. The result is a numerically ef­ficient CO -continuous formulation capable of handling irreversible plastic flow in the presence of softening. Cyclic deformation processes will be briefly discussed as well.

3.1 Yield condition and stabilization

In classical continuum models for quasi-brittle damage, an internal damage variable D is used that progressively reduces the stiffness of the material during the degradation process. A typical constitutive relation in ID reads:

ID: ()" [1 - D (Ii)] Ec: {3.1}

The quasi-brittle damage D is computed directly from a history parameter Ii, which depends on a nonlocal strain e. This constitutive relation clearly shows that the stresses are related to the product of local and nonlocal variables. The efficiency of these models has been illustrated in literature [8]. The equation of nonlocality, which is again a Helmholtz type equation similar to equation (2.27), is coupled to the equilibrium in the zone which experiences damage evolution. However, this format presents no oscillations and a smooth solution is generally obtained. An explanation for this is found, if the diffuse character of the solution of the Helmholtz equation is linked to equilibrium in ID. The diffusion term in the Helmholtz equation causes the following:

e» -+ c:« 1\ e« -+ c:» (3.2)

The product of the nonlocal and the local quantity in (3.1) easily captures a constant stress situation [16].

15

16 Chapter 3 Gradient-enhanced plasticity based on ductile damage

In analogy with quasi-brittle damage models, a ductile damage variable wp is now in­troduced that progressively reduces the yield stress of the material. This leads to a yield condition of the following type:

f (0"",1) = 0- (O") - ag (r,1)

with ay (r) = ayo + h,

K max {1 (7)

0- (O") - a y (r) [1 - wp (K)]

for h> 0

I 0 :S 7 :S t}

(3.3)

(3.4)

(3.5)

where ayo denotes the initial yield strength of the material. It can be noticed that a product of local and nonlocal terms is involved. In order to make a distinction between loading and unloading the Kuhn-Tucker conditions (2.9) apply. With respect to the history parameter an additional set of Kuhn-Tucker conditions can be formulated, analogous to the quasi-brittle damage formulations:

';;2:0, 1-K:S0, ';;{1- K )=0 (3.6)

For the evolution of the plastic damage variable, the following holds:

with O:S wp :S 1 (3.7)

Initially, wp equals 0 and gradually increases towards 1 during plastic deformation due to the increase of the history parameter, which represents the largest nonlocal plastic multiplier that the material has experienced in its loading history {equation (3.5)). Actual equations describ­ing the evolution of the plastic damage will be treated in the next chapter. This formulation

<t enables the gradient-enhanced yield strength a 9 to be constant in the plastic zone, if required.

The stabilizing effect of the algorithm in case of plastic loading is the result of two phe­nomena. First, the local plastic multiplier increases the gradient dependent yield stress ag

with respect to the initial yield strength in the plastic zone (, > 0 1\ h > 0) through the presence of the hardening term in the local yield stress (3.4). The increase of the local plastic multiplier in the plastic zone involves an increase of the nonlocal plastic multiplier through the whole body. This reduces the gradient dependent yield stress according to equation (3.3). The zone of plastic flow may thus extend or narrow during the failure process. For the one­dimensional weakened bar, this reduction will be most present in the plastic zone and will gradually diminish when entering the elastic zone due to the diffusion term in the Helmholtz equation. The net effect is a reduction of the yield strength in the elastic elements near the plastic zone and an increase of the yield strength in the plastic zone with respect to the initial yield strength. Moreover, this algorithm easily captures a constant yield stress in the plastic zone in a similar fashion as gradient-enhanced quasi-brittle damage formulations.

3.2 Rate constitutive equations

In order to enable the discretization of the weak formulation of the governing equations within an incremental-iterative finite element procedure, a number of variables have to be expressed in terms of the degrees of freedom. For the sake of convenience the rate formulation for these variables is derived prior to the weak formulation and discretization.

Chapter 3 Gradient-en1hanced plasticity based on ductile damage 17

In this formulation the nonlocal plastic multiplier and displacements are adopted as the degrees of freedom. Therefore the rate of the local plastic multiplier has to be related to the strain rate and the rate of the nonlocal plastic multiplier. The consistency condition can be used:

o (3.8)

Using the definitions (3.3) - (3.5) the following partial derivatives can be derived:

of oii =N (3.9) = ou ou of - oay [1 - W (K)] = -h[l-wp] (3.10) 0'Y

- o'Y p of of oWp OK OWp OK

(3.11) = [ayo + h'Y] OK 01 01 oWp OK 01

where?;: depends on the material-dependent softening behaviour and Z; is the plastic dam­age evolution switch, which is determined by (3.5):

OK = {O if 1 < K 01 1 if 1~K

(3.12)

Substituting the partial derivatives (3.9) - {3.11} in the consistency condition (3.8) yields:

j = N : 0- - h [1 - wpb + [ayo + h'Y]~: ~~ 'Y = 0 (3.13)

The return-mapping of the stress (2.1O) in rate form reads:

(3.14)

The rate of the local plastic multiplier is expressed in terms of rates of strain and the nonlocal plastic multiplier combining (3.13) and (3.14):

. _ 1 { .4 .' [ ] OWp OK .!.} 'Y - N: 40: N + h [1 _ wp] N. O. e + ayo + h'Y OK 01 'Y (3.15)

Substituting (3.15) in {3.14}, the rate tensor of the stress can be expressed in terms of the strain rate and nonlocal plastic mUltiplier rate:

(3.16)

where

(3.17)

18 Chapter 3 Gradient-enhanced plasticity based on ductile damage

3.3 Weak formulation of differential equations

The problem is governed by the equilibrium condition and the implicit averaging equation which sets the relation between the local and nonlocal plastic multipliers. For a given body force f and a boundary load ~ this set of equation reads:

(3.18)

where a domain 0 in nn (n 1, 2, or 3) is considered, with a boundary r with a unit outward normal n. The classical boundary condition (1' . n = ton r is used for (3.18) l' while the Neumann boundary condition V1' n = 0 on r is adopted for equation (3.18)2'

In order to recast this set of equations in a weak form using the weighted residuals ap­proach, two spaces of weighting functions are defined:

Iwu E [COr} Iw;y E [CO]}

(3.19)

(3.20)

Multiplying (3.I8h with the vectorial weighting function Wu and (3.I8h with the scalar weighting function W;y, and integrating both equations over the domain yields:

{In Wu . { V . (1' + I} dO = 0

In W;y { 1 £2'\721} dO = In W;y'Y dO (3.21)

Applying the divergence theorem and using (1' (1'c with respect to these equations gives for iteration (j + 1) in an iterative-incremental procedure:

I D VWU}' ,IT "+~:n =. (n wu· f~." dn + i Wu . i'u+" dr

In {W7 'Y(j+1) + e '\7W;y . '\7'Y(j+l)} dO = In W;y 'YUH) dO

(3.22)

Decomposing the stresses and plastic multiplier fields in an iterative manner gives:

(1' (j+1) = (1' (j) + 6(1' A 'Y(j+l) 'Y(j) + 6'Y A 1(Hl) = 1(j) + 61 (3.23)

leading to the following linearized weak formulation for the equilibrium condition:

In [Vwur :6(1' dO = In wu' 1(j+1) dO + Ir wu' ~+l) dr

In [V WU r : (1' (j) dO v Wu E Wu (3.24)

and for the linearized averaging equation:

-In w"{ 6'Y dO + In {w"{ 61 + e2Vw;y . V61} dO =

In w"{ 'Y(j) dO - In {W'Y 1(j) + e2vw"{ . V1(j)} dO

(3.25)

Chapter 3 Gradient-enhanced plasticity based on ductile damage 19

Adopting (3.16) for the iterative stress contribution 0(J' in the equilibrium condition, results in:

In Wu . iu+l) dO + £ Wu . ~j+l) dr -In [Vwu r : (J' (j) dO

(3.26)

The weak formulation of the averaging equation is obtained by substituting (3.15) for the iterative local plastic multiplier increment 0/ in equation (3.25):

3.4 Finite element implementation

3.4.1 Discretization of the weak form of the governing equations

Using a vectorial base ~ in 'RP and the Galerkin approach, the displacement vector i1 and the nonlocal plastic multiplier t and the corresponding weighting functions can be discretized in each element Oe:

1\ (3.28)

1\ W;y I r. = N -w-I He -"(-"(

(3.29)

The columns y and t contain the assembled nodal values of the displacements and the non­local plastic multiplier respectively. The nodal arrays 'lYu and 'lYt represent the nodal values of the corresponding weighting functions. N u and N ry denote the interpolation functions for u and t respectively.

For the purpose of this thesis, all further developments will be carried out in a one­dimensional setting. Such an analysis permits to evaluate the ability of this formulation to simulate ductile failure within a finite element framework. Extension to two- and three­dimensional problems is subject of future research. Discretizing equation (3.26) for the entire domain in n 1 and substituting l = T· ~ and t. = [. ~ leads to

1 wT BT [ E h [1 - wP,(j)] ] B OU dO _ -u-u E + h [1 ] -u -n - wP,(j)

r wT BT [E sign ((J(j») [(JyO + h'(j)] oWp I OK I] N _ 0- dO = 1n -u-u E + h [1 - wp,(i)] Of). (j) ot (j) -'Y '1

(3.30)

In 'lY~B~l(j+l) dQ + £ 'lY~B~t.(j+l) dr - In 'lY~B~(J{j} dO V'I.J!u

20 Chapter 3 Gradient-enhanced plasticity based on ductile damage

where Bu represents the matricial strain operator in accordance with the definition (2.2). Similarly the differential equation (3.27) can be discretized to:

-1 wTNT [ E sign (a(j) ] B 6u dD. + - 'Y -'Y E + h [1 ] -u -n - wp,(j)

61 dD. = (3.31)

Vw­-'Y

where B;y represents the gradient operator for the nonlocal plastic multiplier in matrix form. As the resulting equations (3.30) and (3.31) have to be satisfied for all possible weighting arrays, the final discretized form of the linearized weak form of the governing system of equations for iteration (j + 1) reads:

and

( BT [ E h[1 Wp,(j)]] B dD. 6u in-u E+h[1-wp,(j)] -u -

( BT [E sign (aw) [ayo + h/(j)] oWp I OK I 1 N - dD. 61 = in _u E + h [1 - WP,(j)] OK (j) 87 (j) -'Y

[~xt, (j+1) r _mt, (j)

_ ( NT [ E sign (a(j) ] B dO 61J: + in -'Y E + h [1 wP,(j)]-U

- J"I -mt, (j)

(3.32)

where the common expressions for the internal and external nodal forces are adopted and defining similar expressions for the averaging equation:

l~xt, (Hi) = In B~ [(j+1) dO + £ B~ tU+l) dr (3.34)

(3.35)

Jf _mt, (j) = K'Y l' (j) - (Nr 'Y(j) dD. - in (3.36)

with K'Y = N - N - + f B - B - dD. - 1{ T 2 T } - n -'Y -'Y -'Y-'Y (3.37)

Finally the following global system of equations can be obtained:

[ext, (j+1) -lint, (j) (3.38)

Chapter 3 Gradient-enhanced plasticity based on ductile damage

with K(j)

[ext, (HI)

[ Kf_/) K~_7_; 1 K7U K77 -(j) -(j)

= [ [~xt, (j+1) 1 Q

introducing the sub-matrices:

oq = [ :: 1 J. = [ [~nt, (j) 1 _mt, (j) Jf

_mt, (j)

awp I a", I 1 N - dn a", (j) a1 (j) -,

21

(3.39)

(3.40)

The system of equations described by (3.38) and (3.39) holds for plastic deformation. Pure elastic behaviour has the following consequences for the rate of the local plastic multiplier in (3.15) and the stress rate in (3.16):

(3.41)

(3.42)

which results in the following definitions for the sub-matrices (3.40) in the case of elastic behaviour:

Ktr = lB~EBu dO.

(3.43)

-y 1 T 2 T KJ..) = {N- N- +f B- B-} dO. J n -7 -, -7-7

3.4.2 Consistent tangent stiffness matrix and return-mapping

A full Newton-Raphson incremental-iterative procedure will be adopted to solve the governing set of nonlinear equations. In order to obtain a quadratic convergence, a consistent stiffness matrix is required. A stiffness matrix is considered to be consistent if the relation (3.38) is obtained through an exact linearization of the incremental growth of the internal force vector.

22 Chapter 3 Gradient-enhanced plasticity based on ductile damage

The internal nodal forces on integration point level are determined using the return­mapping algorithm (2.10). Linearizing this equation yields:

(3.44)

The last term on the right hand side of this equation is of importance only for nn with n ;::: 2, as for the one-dimensional case this term vanishes:

N

=} oN

ID --+ sign (0")

iJ2 f "iJ(/'z 00" = 0

(Pf &2 =0

(3.45)

Satisfying equation (3.14) will lead to a consistent tangent stiffness matrix with respect to the determination of the nodal forces based on the displacements r. Starting from equation (3.18)2' a similar approach can be followed for the nodal forces -based on the plas­tic multiplier [\ which is rather straightforward. However, considering the the consistent stiffness matrix the plastic damage evolution switch appears in sub-matrices Kt1 and KZ1 in equation (3.40). The variation of this switch during the iterative process inevitably causes a disturbance of the quadratic convergence scheme.

In order to conclude this chapter, table 3.1 shows an outline of the complete consistent incremental-iterative Newton-Raphson procedure as implemented in a finite element frame­work. One important aspect within this scheme is the return-mapping of the stress to the yield surface within the iterative loop. The stress is calculated using the one-dimensional expression of the return-mapping algorithm (2.10) for iteration (j) during increment (i + I):

O"~~t) = 0"(;) + EL1c~~rl) - E sign (O"&t») L1'Y&t) '" " v

(HI) t7 ep, (j)

(3.46)

The projection to the yield surface is realized by imposing fgt) = 0 when determining L1'Y&t} throughout the iterative procedure. The one-dimensional yield condition (3.3) reads:

f (i+I} _ (j) - I (H1) I [ h (HI)] [1 (HI)] O"(j) - O"yO + I(j) - WP,(j)

[ h (i+1)] [1 O"yO + 'Y(j) W(HI)] = 0 p,(J) (3.47)

Substitution of equation (3.46) into (3.47) and introducing the expansion 'Y(Hl) = 'Y(i) + L1'Y(Hl) yields a relation for the total incremental contribution of the local plastic multiplier:

L1 (HI) _ 1 {(HI)' (HI») [ . h (il] [1 (i+1)] } 'Y(j) - E + h [1 _ (HI)] 0" ep, (j) SIgn O"(j) - O"yO + 'Y - Wp,(j) wP,(j)

(3.48)

Note that during computations O"~~~llj) is used in order to determine the sign of O"&t) which will not disturbe the convergence seriously.

Chapter 3 Gradient-enhanced plasticity based on ductile damage

1. Initialize increment counter i 0 Initialize '1/, 1'0 I> Kinematic boundary conditions

2. Next increment i i + 1 Reset iteration counter j 0

3.

4.

5.

6.

Compute te:r;t I>

Next iteration j = j + 1 Loop over elements

Loop over integration points evaluate /(;-1)

If l(j -1) < 0 (elastic behaviour) : t K uue KU'Y' K'Y u ' K'Y'Y' compu e -(j-l»-(j-l)' -(i-I)' _(i-I) I>

Else (plastic behaviour):

evaluate [7?-] and [g",] I< (i-I) _ _"Y (j-1L_

compute K7;"'-1l' Kf?-l) , Kl"-l» K7?-I) I> End

Close integration point loop Close element loop Globalization to !f..j-l) , apply boundary conditions Solve K'J' 1) 8a ::: I t - I, for 8a I> -. - - _ex, (j) _,nt, (j-1) -

Compute: Y(j) = Y(;-l) + 8y 1(j) ::: 1(i-l) + 81

Loop over elements Loop over integration points

External load, boundary conditions

Equation (3.43)

Equation (3.40)

Equations (3.38) and (3.39)

Compute: (1' ep,(j)

I(n «(1'ep,u), }'(;), 1m) I> Elastic prediction (2.10h I> Yield function (3.3)

If l(i) < 0 (elastic behaviour) (1' = (1' ep,(i)

}'(j) }'(j-l)

Else (plastic behaviour) compute:

23

wp ,(;) I> Ductile damage evolution law (4.1) or (4.3)

[;;:'L) and [~it) Ll},U)

}'(j) = }'U-l) + Ll}' (1' (il

End r and t" -int,U) -int,li)

Close integration point loop Close element loop Globalization to I

->nt,(.i)

7. Check for convergence If not converged :::::} step 3 Else :::::} step 2

I> Damage evolution law and softening switch (3.12)

I> Equation (3.48)

I> Return-mapping (2.10)

I> Equations (3.35) ... (3.37)

Table 3.1: Outline of the numerical implementation of the multiplicative gradient-enhanced plasticity formulation

Chapter 4

Numerical results

In this chapter numerical simulations using the ductile damage-based gradient-enhanced plas­ticity model, elaborated in the previous chapter are performed in order to illustrate the local­ization properties of the model and to demonstrate the capability to describe ductile failure in the presence of strain softening. The interpolation polynomials have been chosen quadratic for the displacements and linear for the nonlocal plastic multiplier.

4.1 Ductile damage evolution law

In order to describe the evolution of the plastic damage parameter wp during deformation a proper evolution law (3.7) is required. Evolution laws for quasi-brittle damage, creep etc. can be easily found in literature, but no such laws are yet available for the present model. Within this exploratory analysis, only two academic evolution laws will be considered. The influence of both the introduced parameters and the hardening modulus will be illustrated.

4.1.1 Linear evolution

A first example is the linear evolution of the plastic damage variable:

W (~) = {~ if ~::; ~c p 1 if~>~c

(4.1)

where ~c is a constant denoting the critical value of the history parameter at which the material is completely damaged due to ductile failure. The linear evolution has a constant slope:

(4.2)

The linear evolution of the plastic damage variable is illustrated in figure 4.1 for the sake of clearness.

In order to illustrate the influence of the parameter ~c and the hardening modulus h, the one-dimensional configuration depicted in figure 2.1 is used without the imperfection (d = 0 m). In figure 4.2 the force-displacement curves for the end of the bar (x = L) are depicted for several values of ~c and h. Both parameters influence the slope of the initial softening stage. However, the final end deflection of the bar is mainly influenced by the parameter ~c·

25

26

3 X 104

2.5-

Z 2~ - i

Numerical results

Linear plastic damage evolution 1.2,---.--,---,.---.---.--...,----,---.--,---,

8'0·6

0.4

K=K c

IC X 10-3

Figure 4.1: Linear evolution of plastic damage parameter wp with "'c = 3.0.10-4

Reaction forces versus displacements 2.5 x 10'

Reaction forces versus displacements

~~~O~.I-~~~~~~~~~~~-~~, 3 x10,.,s x 10'" Displacement [ml

Figure 4.2: Influence of the model parameters "'c and h on the global response of a homogeneous tension bar for the linear evolution of the plastic damage parameter; left: influence of the critical history parameter "'c for h = 6.0 . 109 -/!:I and "'c = 3.0 . 10-4 ,6.0 . 10-4 and 9.0 . 10-4 respectively; right: influence of the hardening modulus h for Kc = 3.0· 10-4 and h 6.0 . 109 ,9.0 . 109 and 1.2 . 1010 -/!:I respectively

Chapter 4 Numerical results 27

4.1.2 Exponential evolution

Another possible evolution law for the ductile damage variable is of the exponential type:

(4.3)

where the parameter {3 is a constant which influences the slope of the evolution curve:

0:: ({3,,..) 13,.. e- f3 Ii (4.4)

The exponential evolution of the plastic damage variable is depicted in figure 4.3.

Exponential plastic damage evolution 1.2,---,..----,..----,..----,..---,..-----,

0.8 .

8"" 0.6 .

~~-~7---L--~~--~-~2~.5--~3

X 10-'

Figure 4.3: Exponential evolution of the plastic damage parameter wp with (3 = 3.0 . 103

Analogous to the linear evolution law, the influence of the parameters 13 and h is il­lustrated using the one-dimensional tension bar without the imperfection. The resulting force-displacement curves for the end-deflection of the bar are depicted in figure 4.4 for sev­eral values of the concerning parameters. Evidently the initial slope of the ductile damage evolution is influenced by both 13 and h. In contrast with the linear damage evolution, the maximum end-deflection of the bar is reached for an infinite displacement and a zero stress level. This can be beneficial for numerical simulations in order to avoid the abrupt change of material behaviour when the material fails (e.g. compared to ,.. = "'c for the linear evolution). The long tail of the exponential curve permits to preserve regularity of the stiffness matrix without influencing the equilibrium. However, a remeshing with a proper adaptation of the element topology may still be required once an element has completely failed and is largely deformed.

4.2 Mesh-independency

A consequence of the loss of well-posedness of the boundary value problem, is the pathological mesh-dependence of the finite element solution for a one-dimensional bar with an imperfection.

28

1.6

1.4

Z 8'.2

&, e ~08 ~

0.6

0.2

Chapter 4

Reaction forces versus displacements

~~--~~~~7'----~'.5~-=~2==~~2.5 Displacement [m] x 10-'

Numerical results

Reaction forces versus displacements

1 1.5 Displacement [m[

2.5 X 10-4

Figure 4.4: Influence of the model parameters /3 and h on the global response of a homogeneous tension bar for the exponential evolution of the plastic damage parameter; left: influence of the parameter /3 for h = 6.0.109 !if.x and /3 = 3.0· 103 ,6.0.103 and 9.0· 103 respectively; right: influence of the hardening modulus h for /3 = 3.0 . 103 and h = 6.0 . 109 ,9.0 . 109 and 1.2 . 1010 ~ respectively

In order to demonstrate the mesh-independence of the solution obtained with the gradient­enhanced plasticity formulation based on ductile failure, the one-dimensional bar with an imperfection of figure 2.1 is used. The general parameter settings for these and all the calculations in the remainder of this chapter are summarized in table 4.1. The resulting force-

Parameter Value

Total length of bar L 0.1 m Length of imperfection d 0.02 m Cross-sectional area A 0.01 m:l

Young's modulus E 2.0 . 1010 !/;z Initial yield stress uyo 2.0· lOti ~2

(weakened zone) 1.8.106 ...f!J:

Hardening modulus h 6.0 . 109 f!tr Length parameter e I 0.005 m

Table 4.1: General parameter settings of weakened tensile bar as depicted in figure 2.1

displacement curves are depicted in figure 4.5 for several numbers of elements (20, 40 and 80 elements). Furthermore, the profiles of the displacements and nonlocal plastic multipliers along the bar are shown in figure 4.6. From these figures it is clear that upon mesh refinement the profiles converge towards a unique solution.

Chapter 4

1.8

1.6

1.4

~ <1)1.2 ~

a:: 1 § ·~0.8 <I)

eo: O.G

Numerical results

Reaction forces versus displacements

oL-~~~--~--~--~--~==~~~~ o u x 10-6

29

Figure 4.5: Influence of element size on the force-displacement curve; respectively 20, 40 and 80 elements are used in combination with an exponential evolution of the plastic damage

Displacements along lbe bar 2.5F-X l~O"'---r_-.--N_o_nl..,.OC_al..:p_la,sti_C _m...,.Ul...;tiP...;li_er.,...a_lo-.:ng:::::th=e=bar::::!:==:::::==;-,

20 slemenls1

40sIemenIs SO_I

0.5 .

Figure 4.6: Mesh-independency of displacements and nonlocal plastic multipliers; left: displacement profiles along the bar; right: profiles of the nonlocal plastic multiplier along the bar

30 Chapter 4 Numerical results

4.3 Analysis of localization behaviour

Besides the required mesh-independency of the finite element solution, it is necessary to fur­ther investigate some mechanical properties of the gradient-enhanced plasticity formulation based on ductile damage. The main goal is to simulate ductile failure of a material, i. e. the progressive evolution of a plastically deformed volume into a crack. The evolution of the strain localization characterizes this failure process, which must be described in a reliable fashion.

The proposed formulation, which incorporates a length parameter, enables the simula­tion of ductile damage evolution up to the moment of ductile fracture, without the loss of well-posedness of the governing equations. This is illustrated using the one-dimensional bar with an imperfection, using 44 elements and an exponential evolution for the ductile damage parameter.

The evolution of the displacements along the bar is depicted in figure 4.7. From this fig-

4 x10..s Displacement along the bar

ool--'!'IO.O~1 ~OJ~'2~O.03~~O.04f?--:C:O.05:::---:C:O.06::---:C:O.O:::-7 -::-':O.08=--O::-':.09~O.1 x[mJ

Evolution of displacements along Ibe bar

Increment number x[m)

Figure 4.7: Evolution of the displacements along the bar using 44 elements for an exponential plastic damage evolution; left: displacement profiles along the bar for several increment numbers; right: evo­lution of the displacement profiles during the computation

ure the localization of the deformation is apparent. The corresponding evolution of the total strain is depicted in figure 4.8. This figure shows the localization of the total strain into one point of the bar, which leads to ductile fracture. During the localization of the strain, the evolution of the nonlocal plastic multiplier (and consequently the plastic damage) gradually expands into the unweakened part of the bar, as depicted in figure 4.9.

The influence of the microstructural and micromechanical events during the ductile failure process, are represented by the length parameter E. An increase of E leads to an increase of the volume that is involved during failure. This is noticed by examining the dependence of the width of the localization zone on this length parameter, as can be seen from figure 4.10 (E = 5.0.10-4 ,1.0.10-3 and 5.0·1O-3m). The left figure shows the considerable change of the global response of the tension bar when varying the length parameter. From the right part it

<u

Chapter 4 Numerical results 31

TOlal strain along the bar Evolution of total strain along the bar

0.Q1

0.009

0.008 .

0.007

0.004 .

0.003

0.002

0.001

0.00 0.07 0.08 0.09 0.1 Increment number x[m)

Figure 4.8: Evolution of total strain along the bar using 44 elements for an exponential plastic damage evolution; left: strain profiles along the bar for several increment numbers; right: evolution of the total strain profiles during the computation

3 X10>"3 Nonloeal plastic multiplier along the bar Evolution of nonloeal plastic multiplier along the bar

2.5

0.06 0.07 0.08 0.09 0.1

xfml

Figure 4.9: Evolution of nonlocal plastic multiplier along the bar using 44 elements for an exponential plastic damage evolution; left: profiles of nonlocal plastic multiplier along the bar for several increment numbers; right: evolution of the nonlocal plastic multiplier profiles during the computation

32 Chapter 4.

Reaction forces versus displacements

0.5 1.5 2 2.5 Displal>!ment Iml

Numerical results

x to" Nonl",: .. l plastic multiplier 3.5F---.--,---.--,--'-,------r--,--.----r---,

3.5 4 x 10..0

Figure 4.10: Influence of length parameter on the localization zone; left: force-displacement curves for different values of the length parameter l; right: influence of l on the width of the localization zone

can be concluded that the width of the localization zone, which is coupled to the width of the zone experiencing ductile damage, increases with an increasing value of the length parameter.

4.4 Cyclic deformation process

Next, the proposed gradient-enhanced plasticity algorithm based on ductile failure is used in repeated tension and compression cycles. The resulting global force-displacement curve of one cycle of the deformation process is depicted in figure 4.11 for the one-dimensional situation. From this figure it is clear that the softening behaviour disappears during elastic

2 x 10' Reaction forces versus displacements

~1

~ o ~ 0.5 S

j

-0.5

..... ~

-1L----------~--------~ o 1 2 Displacement [m] x 10-5

Figure 4.11: Behaviour of the one-dimensional bar during a cyclic deformation process

Numerical results 33

K

K ----------------- --------------- -----------------c

E F

A B t

Figure 4.12: Behaviour of the history parameter for the gradient-enhanced plasticity formulation based on ductile failure, corresponding to the cyclic deformation process in figure 4.11

(un)loading (e.g. path CD) until the absolute value of the yield strength is exceeded. Upon further compression (e.g path DE) the softening behaviour continues. This can be illustrated using the history parameter f'i, which governs the evolution of the ductile damage:

f'i,(t) = max{i(T) I 0 ~ T ~ t} (4.5)

The evolution of the history parameter corresponding to the cyclic deformation process of figure 4.11 is depicted in figure 4.12. Apparently, this formulation adopts an evolution of the plastic damage during the whole cyclic process under plastic loading conditions.

However, the failure behaviour of a material can change during the cyclic process. On micro-structural level this can be the case when closure of micro-voids under compression occurs (e.g. metals). During this process no damage evolution will appear, which initially leads to hardening behaviour. With a slight modification this can be adopted in the present one-dimensional formulation by making the history parameter dependent on an alternative equivalent strain measure ceq:

f'i, = max { Ceq (T) I 0 ~ T ~ t} (4.6)

with Ceq - Ic:PI

During elastic unloading (path CD in figure 4.11) the plastic strain remains constant. Upon compression the plastic strain decreases towards zero (from the moment at which the absolute value of the stress exceeds the yield strength; point D) and becomes negative upon further compression. Accordingly, in case of compressive loading this leads to an initial decrease of Ceq followed by an increase upon further compression. However, the evolution of the ductile damage is delayed until the absolute value of the plastic strain exceeds the previous value of the history parameter.

The consequences for the evolution of the history parameter during the cyclic process are depicted in figure 4.13. The dashed line corresponds to the former failure behaviour. Due to equation (4.6) the path DE' is followed instead of DE. During DD' no evolution of the ductile damage parameter wp will occur and the model will exhibit hardening behaviour. Evidently,

34

K

K --------c

c D

A B

9--- -' F

- -- ~ .,... ... ----

D'

Numerical results

t

Figure 4.13: Evolution of the history parameter for distinct behaviour for tensile and compressive conditions; the dotted line refers to figure 4.12

implementing equation (4.6) leads to a different history dependent behaviour ofthe evolution of ductile damage during the cyclic process (e.g. compare (path Be) with (path DE'). It remains to be checked if the definition of equation (4.6) is thermodynamically admissible and if necessary other solutions for Ceq can be considered as well.

4.5 Discussion

The results of the simulations in this chapter show that it is possible to analyze the softening behaviour of a material with the gradient-enhanced plasticity formulation based on ductile failure, which was presented in the previous chapter. As the boundary value problem remains well-posed during the strain localization it is possible to simulate the evolution of the plastic damage over a certain volume which converges towards a line crack.

The model is also capable of simulating hardening behaviour in the offset of the strain localization. The degree of hardening can be tuned by adjusting the hardening modulus h and the parameters determining the ductile damage evolution "'c or p. The interaction between the hardening behaviour and the adopted softening mechanisms is subject of further investigation. It is not yet clear whether possible restrictions on the parameters exist and how this should be dealt with.

Chapter 5

Conclusions and recommendations

In this thesis a new algorithm for ductile damage is investigated within a continuum me­chanics framework. This model is capable of simulating micro-structural dependent ductile deformations and failure, in the presence of strain localization, as experimentally observed for many engineering materials (e.g. metals).

Classical plasticity formulations cannot describe failure phenomena in a reliable manner, since the boundary value problem gets ill-posed. The nonlocal and gradient-enhanced plastic­ity formulations, which have been developed in the past decade, adopt an internal length-scale which is a measure for the micro-structural and micro-mechanical influence of a material point on its neighbours during plastic flow.

The analysis of some examples of gradient-enhanced plasticity models proposed in litera­ture, reveals a number of mathematical inconveniences and difficulties (e.g. the need for small load-increments to preserve convergence, vanishing local yield strength and a large number of degrees of freedom).

In this thesis first an implicit formulation is proposed in which the yield function depends on the sum of a local term and a function of the nonlocal plastic multiplier. The relation between the nonlocal and corresponding local quantity is described with an additional differ­ential equation of the Helmholtz type. This formulation gives rise to unphysical oscillations in the obtained solution. Using a one-dimensional analysis the presence and the origin of the oscillations can be elucidated. In the plastic zone the nonlocal quantity has to be constant, which is difficult to achieve with the solution of a Helmholtz equation, due to its diffusive character.

A second gradient-enhanced formulation, which is based on a ductile damage variable, is proposed to encompass most of the difficulties in other models. Incorporating a product of terms, which dependent on the nonlocal and local plastic multiplier respectively, in the yield function appears to be successful. The local term governs the hardening behaviour of the model, while the evolution of the ductile damage is coupled to the nonlocal term. The algorithm easily captures a constant yield stress in the plastic region for a one-dimensional problem, and oscillations do not occur. The result of the discretization of the coupled set of differential equations within a finite element framework, is a mixed formulation, where the

35

36 V/bllH/~r;1 5 Conclusions and recommendations

displacements and the nonlocal plastic multipliers act as the unknown degrees of freedom. This procedure enables the use of a CO-continuous discretization.

The one-dimensional numerical simulations also show the ability of this algorithm to simulate ductile failure within a certain volume, which converges towards a line crack. Incor­porating the length parameter keeps the boundary value problem well-posed when structural softening occurs and further calculations remain possible. Furthermore, the length parameter determines the volume in which the localization is present.

The formulation is also capable of simulating cyclic deformation processes. The present formulation however incorporates a similar evolution of the ductile damage during the cyclic process under plastic loading conditions. In order to avoid this, a small modification can be implemented, which postpones the evolution of the plastic damage under compressive load­ing conditions. At the onset of compressive loading the material initially shows hardening behaviour. Although this material behaviour is physically more acceptable, the thermody­namical consistency has to be verified.

Future work is concerned with the determination of admissible values of the parameters which determine the evolution of the plastic damage (Kc and (3) and the hardening behaviour (h). Next, the formulation has to be extended to a multi-dimensional model in order to simulate problems, which are more related to real industrial processes. This also requires the adoption of a plasticity formulation which accounts for large strains and rotations.

The comparison of numerical results with experimental results under various loading con­ditions is undoubtfully another point of interest. Such an analysis provides the required information for the simulation of real processes and gives an indication whether the influence of the length parameter, which is isotropic in the present model, has to be anisotropic. Finally, the identification of the length parameter using mixed numerical-experimental techniques is another field of interest [9].

References

[1] De Borst, R. and Miihlhaus, H.-B. Continuum models for discontinuous media. In Fracture Processes in Concrete, Rock and Ceramics: Proceedings of the International RILEMjESIS Conferece, pages 601-618, Noordwijk, The Netherlands, 1991. London: Chapman & Hall.

[2] De Borst, R. and Miihlhaus, H.-B. Gradient-dependent plasticity: Formulation and algorithmic aspects. Int. J. for Num. Meth. in Eng., 35:521-539, 1992.

[3] De Borst, Rand Pamin, J. Some novel developments in finite element procedures for gradient-dependent plasticity. International journal for numerical methods in engineer­ing, 39:2477-2506, 1996.

[4] De Borst, R, Pamin, J., and Geers, M.G.D. On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. European Journal of Mechanics, 1999. Accepted.

[5] De Vree, J.H.P., Brekelmans, W.A.M., and Van Gils, M.A.J. Comparison of nonlocal approaches in continuum damage mechanics. In Computers and structures: an interna­tional journal, volume 55, pages 581-588, 1995.

[6] Geers, M.G.D. Experimental Analysis and Computational Modelling of Damage and Fracture. PhD thesis, Eindhoven University of Technology, The Netherlands, 1997.

[7] Geers, M.G.D., Brekelmans, W.A.M., and De Borst, R Viscous regularization of strain localization for damaging materials: viscous and rate-dependent constitutive models. In G.M.A. Kusters and M.A.N. Hendriks, editors; DIANA Computational Mechanics '94, pages 127-138. Kluwer Academic Publishers, 1994.

[8J Geers, M.G.D., De Borst, R, Brekelmans, W.A.M., and Peeriings, RH.J. Strain-based transient-gradient damage model for failure analysis. Computer methods in applied me­chanics and engineering, 160(1-2):133-154, 1996.

[9] Geers, M.G.D., De Borst, R, Brekelmans, W.A.M., and Peeriings, RH.J. Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene. International Journal of Solids and Structures, 1998. Accepted.

[10] Hill, R A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids, 6:236-249, 1958.

37

38 References

[11] Meftah, F., Pijaudier-Cabot, G., and Reynouard, J.M. A CO Finite Element in Gradient Plasticity for Localized Failure Modes Analysis. ConJ. paper of fourth World Congres on Compo Mech., pages 1-13, 1998.

[12] Mikkelsen, L.P. Post-necking behaviour modelled by a gradient dependent plasticity theory. In Int. J. for Solids and Structures, volume 34, pages 4531-4546, Great Britain, 1997. Pergamon Press.

[13] Miihlhaus, H.-B. and Aifantis, E.C. A variational principle for gradient plasticity. In International Journal of Solids and Structures, volume 28, pages 845-858, Great Britain, 1991. Pergamon Press.

[14] Nix, W.D. and Gao, H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of the mechanics and physics of solids, 46:411-426, 1998.

[15] Pamin, J.K. Gradient-Dependent Plasticity in Numerical Simulation of Localization Phe­nomena. PhD thesis, Delft University of Technology, The Netherlands, 1994.

[16] Peerlings, RH.J., de Borst, R, Brekelmans, W.A.M., and de Vree, J.H.P. Gradient enhanced damage for quasi-brittle materials. In International Journal for Numerical Methods in Engineering, volume 39, pages 3391-3403, 1996.

[17] Pietruszczak, S. and Mr6z, Z. Finite element analysis of deformation of strain softening materials. In International Journal for Numerical Methods in Engineering, volume 17, pages 327-334, 1981.

[18] Polizzotto, C., Borino, G., and Fuschi, P. A thermodynamically consistent formulation of nonlocal and gradient plasticity. In Mechanics Research Communications, volume 25, pages 75-82, USA, 1998. Elsevier Science Ltd.

[19] Sluys, L.J. and De Borst, R Solution methods for localization in fracture dynamics. In Fracture Processes in Concrete, Rock and Ceramics: Proceedings of the International RILEM/ESIS Conferece, pages 661-671, Noordwijk, The Netherlands, 1991. London: Chapman & Hall.

[20] Steinmann, P. and Willam, K. Localization within the framework of micropolar elasto­plasticity. In Advances in continuum mechanics, pages 296-313, Berlin, 1991. Springer Verlag.

[21] Stromberg, L. and Ristinmaa, M. FE-formulation of a nonlocal plasticity theory. Comput. Methods Appl. Mech. Engrg., 136:127-144, 1996.

[22] Svedberg, T. and Runesson, K. A thermodynamically consistent theory of gradient­regularized plasticity coupled to damage. In International Journal of Plasticity, vol­ume 13, pages 669-696, Great Britain, 1997. Pergamon Press.

[23) Wang, W. Stationary and propagative instabilities in metals - a computational point of view. PhD thesis, Delft University of Technology, The Netherlands, 1997.

[24] Wang, X.C. and Habraken, A.M. An elasto-visco-plastic damage model: from theory to application. Journal de physique, 6, 1996.

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[25] Willam, KJ. and Etse, G. Failure assessment of the extended Leon model for plain con­crete. In Proc. Int. Conf. Computer Aided Analysis and Design of Concrete Structures, pages 33-70, Swansea, 1990. Pineridge Press.

[26] Xikui, Li and Cescotto, S. Finite element method for gradient plasticity at large strains. International journal for Numerical Methods in Engineering, 39:619-634, 1996.