A MULTIFIELD VARIATIONAL FE APPROACH FOR NONLOCAL DAMAGE

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.) R. Owen and M. Mikkola (assoc. eds.) Jyväskylä, July 2004 A MULTIFIELD ARIATIONAL FE APPROACH FOR NONLOCAL DAMAGE Elena Benvenuti, Benjamin Loret, and Antonio Tralli Department of Engineering Ferrara I-44100, University of Ferrara ebenvenuti@ing.unife.it Laboratoire Sols, Solides, Structures P.O. Box 1100, F Grenoble Cedex 9 Universite J. Fourier, France s:loret@hmg.inpg.fr,atralli@ing.unife.it Key words: multifield formulations, nonlocal damage, finite elements Abstract. A multifield finite element model is presented for isotropic damaging materials with strain-driven damage. The nonlocal equivalent strain definition is inserted into the work function by introducing a Lagrangian multiplier. This Lagrangian formulation depends on: displacement, nonlocal equivalent strain and Lagrangian multiplier, and provides a variational basis supporting multifield formulations of both integral and implicit gradient type. Finite element results are presented for some representative two-dimensional examples. 1

2 1 INTRODUCTION Elastodamaging materials can be conveniently modelled by means of softening stressstrain laws of damaging type and simulating the damage evolution as a strain-driven process. The presence of a softening branch in the stress-strain law induces pathological effects. For instance, well-posedness of the boundary value problem is lost, while the energy dissipation vanishes over the process zone where the strain localizes. In finite element models, the strain localization zone coincides with the smallest mesh size originating a pathological meshdependency of the numerical results. Mesh-dependency can be alleviated if the constitutive equations are enriched by an internal length. For instance, nonlocal formulations introduce a characteristic length by having recourse to integral averages or gradient expansion of certain variables [1][2][3][4] [5] [6][7] [8][9]. This allows to restore the mesh independency of the finite-element results, preserving a finite size of localization band. In [10], the presence of a (plastic) internal length is shown to allow the mechanical information to travel some distance, precluding various instabilities. In a dynamic context, alternatives are available, as a Duvaut-Lions viscoplasticity produces in the field equations a length as a product of an elastic wave-speed times the relaxation time [11]. The purpose of the present contribution is to develop a general multifield formulation for nonlocal definitions of the equivalent strain of both integral type and implicit gradient type. The starting constitutive assumption is that damage at a point depends on the total strain history. After integration of the mechanical work along a radial strain path, the increment of the mechanical work is written as a strain function [12][13][14][15]. The nonlocal strain function is then treated as a constraint and multiplied by a Lagrangian multiplier [16]. In this form, it is added to the increment of the mechanical work previously obtained. The resulting Lagrangian formulation depends not only on the displacement field but also on the nonlocal equivalent strain and the Lagrangian multiplier. The discrete form of the boundary value problem associated to the Lagrangian functional is analyzed through the finite element method. The Lagrangian formulation provides a unified framework encompassing both integral and implicit gradient definitions of the nonlocal equivalent strain: a switch from the former to the latter is possible at the expense of changing few terms of the incremental-iterative formulation [16]. Finite element results are presented for the implicit gradient model in a one-dimensional and two-dimensional case. 2 Nonlocal damage In the context of nonlocal formulations for elasto-damaging materials, the damage parameter is usually governed by a number of nonlocal definitions of the equivalent strain [7]. In particular, taking as nonlocal a strain function makes it possible to solve the loadingunloading conditions at each Gauss point of the discretization in a single step. For instance, a nonlocal integral equivalent strain field { ɛ} can be obtained by an integral weighted average 2

3 of the local field ɛ over the body : { ɛ}(x) = 1 r (x) α(x, y) ɛ(y) dy. (1) In Eq. (1), r (x), the reference volume at point x, is defined as r (x) = α(x, y) dy, so preserving homogeneous fields. The space function α(x, y) is a weighting function, which kills informations coming from points y located at a distance from x larger than a characteristic length l. For instance, the exponential Gauss function e k2 x y 2 /l 2 is equal to e k2 on the sphere of radius l centered at x, where k is a parameter. First proposed in [2], implicit gradient relationships can be obtained by expanding in Taylor series the integrand of (1) under the hypothesis of symmetric integration domain. The nonlocal variable { ɛ} is then approximated through the implicit gradient differential equation at a point x { ɛ}(x) = ɛ(x) + c 2 { ɛ}(x), (2) where 2 is the Laplacian operator. The positive scalar c determines the width of the nonlocal effect and it has the dimension of a squared length. Its relationship with the characteristic length l of the Gauss weighting function of the integral form (1) has been also discussed in [4]. To solve the differential equation (2), the boundary condition { ɛ}(x) n = 0 on the body surface is usually imposed, reminiscent of the one obtained for gradient plasticity in [17]. In this context, multifield formulations have been proposed, where the nonlocal definition of the equivalent strain is assumed as an independent variable: the Neumann problem associated to Eq. (2) endowed with the above boundary condition is solved together with the equilibrium equation. These multifield formulations based on implicit gradient formulations have been proved computationally more stable and suitable for modelling fracture and localization problems than other (explicit) gradient techniques [4], [5]. Moreover, approximating { ɛ} by Eq. (2) as an independent field limits the occurrence of oscillations of the computed stress profiles. Let us denote R the nonlocal operator: R(ɛ) = { ɛ}(ɛ). (3) Special cases of the function R considered in the sections below are 1 α(x, y) ɛ(ɛ(y)) dy, integral case, R(ɛ)(x) = r (x) ɛ(ɛ(x)) + c 2 { ɛ}(ɛ(x)), +B.C., implicit gradient case. at any point x of the volume. The nonlocal equivalent strain is expressed as a strain function by Eq. (3) and controls the damage evolution through the loading-unloading conditions: g = { ɛ} κ, ω 0, g ω = 0. (5) Now the damage threshold κ represents the highest value the nonlocal equivalent strain has ever reached during the entire loading history, (4) κ n+1 = sup{ ɛ}(t). (6) t τ 3

4 3 Incremental problem Let us consider a loading history within the interval of interest τ = [0, T ], and discretize this interval by a time-like parameter t into N nonoverlapping intervals [18]. At a generic initial time t n and at each material point, the displacement field u n, the isotropic damage parameter ω n, and the Cauchy stress σ n are known. The strain field ɛ n is assumed equal to s u n, the symmetric part of the displacement gradient. After a load increment over the time step [t n, t n+1 ], the material response at t n+1 has to be obtained, namely (u n+1, ɛ n+1, ω n+1, σ n+1 ). Suppose that, for instance solving equilibrium, the (iterative) value u n+1 is given, so that the strain ɛ n+1 and, consequently, the local equivalent strain ɛ n+1 are known. Because both damage parameter and damage threshold are non-decreasing during the loading history, it can be shown that the loading-unloading conditions can be summarized as [15]: g n+1 = { ɛ} n+1 κ n+1 0, ω g n+1 = 0, ω n+1 = max { ω n, ω({ ɛ} n+1 ) }, { ɛ} n+1 = R(ɛ n+1 ). (7) The stress-strain law at time t n+1 σ n+1 = ( 1 ω n+1 ) Eɛn+1 (8) and the loading-unloading conditions (7) delineate the actual constitutive framework. Let the elasto-damaging body of volume be subjected to volume forces b n, to boundary tractions p n on the free part f of its boundary and to prescribed displacement ū n on the constrained part c of. During the time step [t n, t n+1 ], the volume forces and the surface forces increase by b, and p, respectively. Given (u n, ω n, σ n ), the corresponding values at the instant t n+1, (u n+1, ω n+1, σ n+1 ), can be found by solving: divσ n+1 + b n+1 = 0 in, (9a) σ n+1 n = p n+1 on f, u n+1 = ū n+1 on c, (9b) { ɛ} n+1 = R(ɛ n+1 ) in, subject to the stress-strain law (8) and to the loading unloading conditions (7). In (9b), n represents the vector normal to the boundary. 4 Multifield formulation Consistently with the hypothesis of strain-driven damage, the damage is governed by the loading-unloading conditions (7) and the nonlocal relationship (3) expressing { ɛ} n+1 as a function of ɛ. After integration of the mechanical work along a strain radial path stemming from the strain state ɛ n to the strain state ɛ n+1, the finite increment of the mechanical work writes[15][16] ψ(ɛ n+1, H) = φ(ɛ n+1 ) + γ(ɛ n+1, H), (10) (9c) 4

5 where φ(ɛ n+1 ) = ( )1 1 ω n+1 2 ɛ n+1 Eɛ n+1 ( )1 1 ω n 2 ɛ n Eɛ n (11) is path-independent and 1 dω(ɛ) γ(ɛ n+1, H) = ɛ Eɛ dɛ, 2 dɛ γ(ɛ n+1, H) 0 (12) H depends on the final strain ɛ n+1 and on the strain path H. The complete derivation is not reported for brevity of exposition (see [15]). In a compact notation, according to the loading-unloading conditions (7), the damage derivative is written dω = d dɛ ω d{ ɛ}, where the dɛ scalar 0 if κ = κ n, d ω = ω κ otherwise (13) is introduced. The total work function at the time t n+1 is obtained by integrating over the volume the work density function Ψ(ɛ n+1 ) given by Eq. (10) and subtracting the work of the surface forces p n+1 and the body forces b n+1 : P(u n+1 ) = Ψ(ɛ n+1 ) d p n+1 u n+1 ds b n+1 u n+1 d, (14) f where ɛ n+1 = s u n+1. Let us rewrite equation (3) as a constraint between the nonlocal field { ɛ} n+1 and the strain field ɛ n+1 : G(ɛ n+1, { ɛ} n+1 ) = { ɛ} n+1 R(ɛ n+1 ) = 0 in. (15) Assume further that the field { ɛ} n+1 is an additional independent variable. The displacement field u n+1 that realizes the stationarity of P(u n+1 ) (14) is obtained by solving the constrained stationarity problem: u P(u n+1 ) = 0 (16) subject to the loading-unloading conditions (7) and to the constraint (15). The first key choice is assuming the nonlocal equivalent strain { ɛ} as an additional independent field. That makes it possible to consider the two-field functional ( P(u n+1, { ɛ} n+1 ) 1 ω({ ɛ}n+1 ) ) 1 2 ɛ ( )1 n+1 Eɛ n+1 d 1 ωn 2 ɛ n Eɛ n d + + Ψ n d + ɛn+1 ɛ n 1 2 ɛ Eɛ dω dɛ dɛ d b n+1 u n+1 d p n+1 u n+1 ds. f The second key choice is to introduce a total work function embodying the constraint (15) multiplied by a Lagrangian multiplier λ n+1 : L(u n+1, { ɛ} n+1, λ n+1 ) = P(u n+1, { ɛ} n+1 ) + λ n+1 G(ɛ n+1, { ɛ} n+1 ) d, (18) 5 (17)

6 where ɛ follows a radial strain path. The Lagrangian functional (18) depends on three independent fields, (u n+1, { ɛ} n+1, λ n+1 ). The above constrained stationarity problem is equivalent to find (u n+1, { ɛ} n+1, λ n+1 ) that realize the stationarity of the Lagrangian functional (18) subject to the loading-unloading conditions (7). 5 Finite element formulation {u,{ ɛ},λ} L(u n+1, { ɛ} n+1, λ n+1 ) = 0 (19) Let us subdivide the body into N e finite elements and approximate the unknown fields over each finite element through the nodal components U n+1, {Ē} n+1, and Λ n+1 by means of the interpolation functions N u (x), N { ɛ} (x), N λ (x): u n+1 (x) = N u (x)u n+1, { ɛ} n+1 (x) = N { ɛ} (x){ē} n+1, λ n+1 (x) = N λ (x)λ n+1, (20) and the associated (symmetrized) gradients: ɛ n+1 (x) = B u (x)u n+1, ɛ n+1 (x) = B { ɛ} (x){ē} n+1, λ n+1 (x) = B λ (x)λ n+1. (21) Approximation of the Lagrangian functional (18) through the discrete fields (20) and (21) leads to the Lagrangian function [15] 1( ) L(U n+1, {Ē} n+1, Λ n+1 ) = L n + 1 ωn+1 U t 2 n+1 B t ueb u U n+1 d + ) + Λn+1( t N t λ N { ɛ} + B t λcb { ɛ} {Ē}n+1 d N λ Λ n+1 f(u n+1 ) d + where + Un+1 {Ē} D n+1 ω U 1 U n 2 Ut B t ueb u U du d b n+1 N u U n+1 d p n+1 N u U n+1 d, f L n L(U n, {Ē} n, Λ n ) = Ψ n d (22) 1( ) 1 ωn U t 2 n B t ueb u U n d. (23) The structure of the Lagrangian function L(U n+1, {Ē} n+1, Λ n+1 ) is general, because it encompasses both integral and implicit gradient constraints. In particular, the integral constraint (4), at a prescribed Gauss point x g within the set of N g Gauss points of the finite element mesh, writes N g R( ɛ n+1 )(x g ) = α gj ɛ n+1 (x j ), (24) j=1 6

7 where α gj = e k2 xg x j 2 l 2 denotes the gj component of the second-order weight matrix α (see also [9]). The local equivalent strain ɛ n+1 is a known function of the strain field and consequently of the displacement nodal vector U n+1, so Eq. (24) becomes the displacement function N g f(u n+1 )(x g ) = α gj ɛ j (U n+1 )(x j ). (25) j=1 Moreover, in the case of integral constraint, the gradient parameter c must be set equal to zero in the Lagrangian function (22). When the implicit gradient constraint is adopted, the function f(u n+1 ) coincides with the local equivalent strain definition: f(u n+1 ) = ɛ(u n+1 ), (26) and the parameter c is different from zero in Eq.(22). The derivative of the damage variable with respect to the equivalent strain has been indicated for brevity by 0 if κ = κ n, D ω = ω κ N (27) { ɛ} otherwise, where, by the chain rule, N { ɛ} = κ {Ē}. Note that the factor D ω results to be discontinuous at incipient damage. The stationarity points of the function L n+1 L(U n+1, {Ē} n+1, Λ n+1 ) (22) are obtained by solving its Euler Lagrange equations. Because the Euler-Lagrange equations are nonlinear, an iterative Newton-Raphson procedure is activated based on their linearization. At iteration i + 1, an approximation of the unknowns χ i+1 n+1, such that χ i+1 n+1 (U n+1, {Ē} n+1, Λ n+1 ), is obtained by solving The matrices and vectors in system (29) in extenso write: K uu K u{ ɛ} K uλ K i n+1 = K u{ ɛ} K { ɛ}{ ɛ} K { ɛ}λ K t uλ K t { ɛ}λ 0 i χ i+1 n+1 = χ i n+1 + δχ i+1 (28) K i n+1δχ i+1 = R i n+1. (29) n+1 δu, δχ i+1 = δ{ē} δλ i+1 n+1, i L/ U R i n+1 = L/ {Ē} L/ Λ n+1. (30) 7

8 The iterative tangent matrix K i n+1 is symmetric and has components: ( ) K uu = 1 ωn+1 B t dk d (U n+1 ) d 2 f(u n+1 ) u EB u d + d N λ Λ n+1 d, (31a) du n+1 du 2 n+1 K u{ ɛ} = B t ueb u U n+1 (D ω) n+1 d, (31b) (df(u n+1 )) tnλ K uλ = d, (31c) du n+1 1 K { ɛ}{ ɛ} = (D ω) n+1 2 Ut n+1b u EB u U n+1 d, (31d) ( ) K { ɛ}λ = N t { ɛ} N λ + B t { ɛ}cb λ d. (31e) In K uu (31a), it has been set K d (U n+1 ) = (D ω) n+1 {Ē} n+1 U n Ut n+1b t ueb u U n+1, (32) The matrix (D ω) n+1 in Eq. (31d) collects the second derivatives of the damage variable with respect to the nonlocal equivalent strain at t n+1 : with N t { ɛ} N { ɛ} = 2 κ n+1 {Ē} 2 n+1 0 if κ n+1 = κ n, (D ω) n+1 = ω n+1 N t κ { ɛ}n 2 { ɛ} n+1 otherwise, (33). Analogously to (D ω) n+1, (D ω) n+1 turns out being discontinuous at incipient damage and significantly increases as soon as the damage process is activated. As above remarked, the parameter c and the function f(u n+1 ) change according to the nonlocal relationship which is adopted. These quantities appear in the residuals ( L/ U) i n+1 and ( L/ Λ) i n+1, and in the expressions of the the submatrices K uu and K uλ. All the other submatrices of the tangent matrix are f-independent. 6 Iterative-incremental procedure A mixed load-displacement loading control (arc-length) has been adopted. The arc-length algorithm of the generalized displacement control (GDC) has been used [19]. In the GDC, the iterative increment of the arc-length parameter δµ i+1 multiplying the load increment P at the iteration i + 1 depends on degrees of freedom moving with monotonically increasing displacements. The GDC has been proposed for displacement-based formulations, and later extended to multifield formulation by [20]. Here, following [19], the generalized displacement control formula has been modified to account for the presence of multiple fields. To simplify 8

9 notation, the n + 1 index is omitted. The iterative increment of the unknowns δχ i+1 is split into δχ i+1 = δχ i+1 1 +δµ i+1 δχ i+1 2, (34) where δµ i+1 denotes the arc-length parameter at the iteration i + 1 and P δχ i+1 1 = K i 1 R i, δχ i+1 2 = K i 1 0. (35) 0 From Eqs. (35), because δχ i+1 = (δu i+1, δ{ē} i+1, δλ i+1 ), we get the expressions of the displacement increments δu i+1 1 = ( i K 1 uu K 1 ) R u i uλ, δu i+1 2 = ( K 1 uu K 1 ) P i uλ. (36) K 1 u{ ɛ} R ɛ R λ K 1 u{ ɛ} Finally, the expression of the arc-length parameter of the GDC for the present multifield formulation writes ( ) U i T δu i+1 δµ i+1 ( U i, δµ i 1 ) = ( ). (37) U i T δu i+1 2 Eq. (37) makes it possible to move along the tangent direction at any Newton-Raphson iteration. This becomes important in all the cases where an error over the choice of the direction of the iterative Jacobian matrix can cause divergence of the numerical scheme. Tests have been carried out also using monotonically increasing degrees of freedom of the nonlocal equivalent strain and the Lagrangian multiplier. However, the best performance was found with formula (37). This depends on the fact that, unlike damage, the Lagrangian multiplier does not monotonically evolve. It can increase at the initial stage of the damage process up to a peak and finally decreases, because it is proportional to the damage derivative. The numerical robustness of the three-field model is influenced by the condition number of the iterative equilibrium matrix K i defined by Eqs. (30) and (31). It has been here observed that, in the presence of damage localization, the condition number of the iterative matrix can reach very high values, thus penalizing the robustness of the iterative-incremental procedure. To this purpose, the iterative tangent matrix i K uu Ku{ ɛ} K uλ K i n = K t u{ ɛ} K { ɛ}{ ɛ} K { ɛ}λ (38) K t uλ K t { ɛ}λ 0 has been here used throughout the calculations, where the terms containing the first damage derivative have been dropped out by assuming ( ) K uu = 1 ωn+1 B t u EB u d, (39) n+1 K u{ ɛ} = 0. (40) 0 0 9

10 Nevertheless, the necessity of calculating the first and second damage derivatives emerges at each equilibrium iteration, because they enter the calculation of the residuals and of the submatrix K { ɛ}{ ɛ} (31d). The present numerical tests have shown that the rate of convergence is not significantly altered by assumptions (39). The computational cost of the three field formulation is larger than the cost of the two-field model by Peerlings et al.[2], due to the presence of an additional independent field, the Lagrangian multiplier. However, because the iterative tangent matrix is symmetric, the storage of 5 sub-matrices is required, while in the two-field model it is necessary to store 4 sub-matrices. All finite element codes have been entirely implemented by the authors on a Matlab 5.3 platform and run on a Pentium I 2.4 GHz Intel processor. According to the present experience, the CPU-times required for the three and the two-field model are comparable, probably because Matlab is not a compiled language. 7 Examples Finite element results are presented for the three-field function obtained in case of implicit gradient definition of the nonlocal equivalent strain. A two-dimensional notched specimen subject to tensile loading and a plate under a compression loading are analyzed. Mesh independence of the results is checked and a comparison with the implicit gradient twofield model is carried out. It is shown that the three field and the two-field models lead to load-displacement profiles which are comparable. The maps of damage, nonlocal equivalent strain and Lagrangian multiplier are reported showing their localization. In particular, the Lagrangian multiplier, proportional to the damage derivative, instantaneously increases from zero up to several orders of magnitude at the beginning of the damage process, and subsequently decreases. In the numerical examples, eight-noded serendipity elements approximate the displacement field, whereas four-noded bilinear elements are chosen for the equivalent strain field and the Lagrangian multiplier field. This choice stems from the necessity of comparing the present results with the ones obtained with the two-field model of implicit gradient type [2]. A grid of four Gauss points per element has been used. 7.1 Two-dimensional tensile specimen The concrete notched specimen in Figure 1 has been experimentally studied by [21], and is a rather common benchmark test to compare numerical and experimental results (e.g. [6]). In the experiment of Hassanzadeh, the specimen has been first loaded under deformation control in pure tension up to the peak load, and then subjected to a constant ratio between normal and tangential displacement. In this numerical example, a uniform distribution of traction load has been considered applied at the top of the specimen, whereas the nodes at the bottom are fixed. The geometry has been studied in a plane stress condition. Due to the symmetry of the geometry under tensile loading, only one half of the specimen has been considered. The material has been modelled setting the Young s modulus E=32900 MPa, 10

11 Figure 1: Geometry of the tensile specimen (length unit [mm]) Poisson s ratio ν = 0.2. The damage evolution law of exponential type { 0 if κ n < κ i, ω n = ( 1 ) κ i κ n 1 α + αe β(κ n κ i ) otherwise (41) has been adopted, where κ i = , α = 0.96 and β = 350. The equivalent strain is defined by: r 1 ɛ(ɛ) = 2r(1 2ν) I 1(ɛ) + 1 (r 1) 2 2r 2r (1 2ν) 2 I2 1(ɛ) + (1 + ν) J 2(ɛ), (42) 2 where I 1 = tr(ɛ) and J 2 = 3tr(ɛ 2 ) tr 2 (ɛ) are the strain tensor invariants [20]. The load-displacement results obtained on a coarse mesh, a medium and a fine mesh have been proven almost mesh independent (Figure 2). The meshes are displayed in Figure 3. The damage, the nonlocal equivalent strain and the Lagrangian multiplier localize within the zone of the notch. Their contour plots at the end of the loading process are shown in Figures 3a-b-c for each mesh. It can be observed that the adoption of the gradient constraint keeps finite the size of the localization zones. 7.2 Plate under compression Following an analogous example studied, for instance, by Pamin in gradient plasticity [3], the plane stress two-dimensional plate in Figure 4 has been analyzed having basis b=60 mm and height h=120 mm. The plate is clamped at the bottom and subject to a uniform compressive force applied at the upper side. The damage evolution of exponential type (41) has been adopted assuming k i = , α = 0.96 and β = 350. The equivalent strain definition ɛ(ɛ) = 3 [ɛ i ] 2 (43) 11 i=1

12 70 60 P [Kgf/mm] u [mm] Figure 2: Load per unit thickness versus vertical displacement of the top of the tensile specimen obtained for the coarse (continuous line), the medium (dashed line) and the fine meshes (dash-dotted line) by using the three-field Lagrangian model (c=1 mm 2 ) has been exploited. The dark zone placed at the left bottom corner in Figure 4 is characterized by a 10 % smaller damage threshold k i. Two meshes made of 6 by 15 and 12 by 15 finite elements have been used for the simulations. The coarse mesh can be seen in Figure 4. The material is characterized by Young s modulus E = MPa and Poisson s ratio ν = The gradient parameter c has been set equal to 1 mm 2. The vertical displacement of point B at the top of plate has been plotted versus the resultant of the loads for the two meshes (Figure 5a). As shown in Figure 5b, the load-displacement responses given by the three-field (continuous line) and the two-field model (dashed line) are almost coincident. Damage localizes along a 45 o band from the predamaged left bottom corner. The same localization can be observed for the nonlocal equivalent strain and the Lagrangian multiplier. Figures 6a-b-c display the evolution of ω, { ɛ} and λ, respectively, corresponding to the states labelled with 1, 2, 3 in Figure 5 along the load-displacement curve of the fine mesh. Again, the action of the Lagrangian multiplier increases at the beginning of the damaging process and decreases thereafter (Figure 6c). Correspondingly, the localization band of the pressure-like term first enlarges and then progressively shrinks for increasing values of the top displacement. That is a consequence of the fact that λ is proportional to the damage derivative. 8 Conclusions Starting from the hypothesis of isotropic strain-driven damage, a work function has been obtained by integration of the mechanical work along a strain radial path. The definition of nonlocal equivalent strain of both integral and implicit-gradient type has been treated 12

13 Figure 3: Profiles obtained for the two-dimensional tensile notched specimen with the coarse, the medium and the fine meshes at the end of the loading process (c=1 mm 2 ): damage (a), nonlocal equivalent strain (b), Lagrangian multiplier (c). as a constraint, and subsequently embodied by a Lagrangian multiplier into the total work function. So a three-field formulation has been constructed, depending on: displacement, nonlocal equivalent strain and Lagrangian multiplier. Finite element tests have been carried out for tensile and compressive loadings on some meaningful one-dimensional and twodimensional geometries assuming an implicit gradient type formulation. A more detailed derivation of the present Lagrangian formulation can be found in [16]. It has been there shown that the boundary condition of the implicit gradient constraint can be obtained via variational arguments. Moreover, in [16], the connections of the Lagrangian formulation with the two-field implicit gradient model by [2] have been widely highlighted. 13

14 B Figure 4: Geometry of the compressed plate with the 6 15 mesh (length unit [mm]) P [Kgf/mm] P [Kgf/mm] u [mm] u [mm] Figure 5: Load per unit thickness versus vertical displacement of the top of the compressed plate obtained with the 6 15 mesh (dashed line) and the mesh (continuous line) (c = 1 mm 2 )(a). Load per unit thickness versus vertical displacement of the top of the compressed plate obtained with the Lagrangian three-field model (continuous line) and the two-field model (dashed line) (c = 1 mm 2 )(b). REFERENCES [1] G. Pijaudier-Cabot, Z. Bažant. Nonlocal damage theory. J. Engng. Mech., 10, , [2] R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans, J.H.P. de ree. Gradient enhanced damage for quasi-brittle material. Int. J. Num. Meth. Engng., 39, , [3] J. Pamin. Gradient-dependent Plasticity in Numerical Simulation of Localization Phenomena. PhD Thesis, Delft University,

15 Figure 6: Evolution of the damage, the nonlocal equivalent strain, and the Lagrangian multiplier for the compressed plate in the fine mesh at the points 1,2,3 displayed in Figure 5a (c = 1 mm 2 ). [4] R.H.J. Peerlings, M.G.D. Geers, R. de Borst, W.A.M. Brekelmans., A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Structures, 38, , [5] A. Simone, H. Askes, J.L. Sluys. Incorrect initiation and propagation of failure in nonlocal and gradient-enhanced media. Int. J. Solids Structures, 41, , [6] C. Comi. A non-local model with tension and compression damage mechanisms. Eur. J. Mech. A/Solids, 20, 1-22, [7] M. Jirásek, B. Patzák. Consistent tangent stiffness for nonlocal damage models. Comp. Struct., 80, , [8] E. Benvenuti, G. Borino, A. Tralli. A thermodynamically consistent nonlocal formulation for damaging materials. Eur. J. Mech. A/Solids, 21, ,

16 [9] E. Benvenuti, A. Tralli. Iterative LCP solvers for nonlocal loading unloading conditions. Int. J. Num. Meth. Engng., 58(15), , [10] B. Loret, F.M.F. Simoes, J.A.C. Martins. Regularization of flutter ill-posedness in fluid saturated porous media. Continuum Thermomechanics, G. Maugin Ed., Kluwer, Dordrecht, [11] B. Loret, J.H. Prévost. Dynamic strain localization in elasto-visco-plastic solids. Comp. Meth. Appl. Mech. Engng., 83, , [12] P. Carter, J.B. Martin. Work bounding functions for plastic materials. J. Appl. Mech., 98, , [13] M. Ortiz, E.A. Repetto. Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids, 47, , [14] C. Miehe, J. Schotte, M. Lambrecht. Homogenization of inelastic solid materials at finite strains based on incremental principles. Application to the texture analysis of polycrystals. J. Mech. Physics Solids, 50, , [15] E. Benvenuti. Damage integration in the strain space. In press on Int. J. Solids Structures, doi: /j.ijsolstr , [16] E. Benvenuti, B. Loret, A. Tralli. A unified multifield formulation for nonlocal damage. Forthcoming on Eur. J. Mech. A/Solids. [17] H.B. Mühlhaus, E.C. Aifantis. A variational principle for gradient plasticity. Int. J. Solids Structures7, , [18] J.C. Simo, T.J.R. Hughes, Computational inelasticity, Springer erlag., [19] R. de Borst. Nonlinear analysis of frictional materials. PhD thesis, Delft University Technology, [20] M.G.D. Geers. Enhanced solution control for physically andeometrically non-linear problems; Part I- The subplane control approach; Part II- Comparative performance analysis. Int. J. Num. Meth. Engng., 46, , [21] M. Hassanzadeh. Behavior of fracture process zones in concrete influenced by simultaneously applied normal and shear displacements. PhD Thesis, Institute of Technology, Lund,