Interaction of void spacing and material size effect on inter-void flow localisation
I. Holte, A. Srivastava, E. Martínez-Pañeda, C. F. Niordson, K. L. Nielsen
IInteraction of void spacing and material sizeeffect on inter-void flow localisation
Ingrid Holte
Department of Mechanical Engineering
Technical University of Denmark, Kgs. Lyngby, DenmarkE-mail: [email protected]
Ankit Srivastava
Department of Materials Science and Engineering
Texas A&M University, College Station Texas, USAE-mail: [email protected]
Emilio Mart ´ı nez-Pa ˜ n eda Department of Civil and Environmental Engineering
Imperial College, London, UKE-mail: [email protected]
Christian F. Niordson , Kim L. Nielsen Department of Mechanical Engineering
Technical University of Denmark, Kgs. Lyngby, Denmark E-mail: [email protected] E-mail: [email protected]
The ductile fracture process in porous metals due togrowth and coalescence of micron scale voids is not onlyaffected by the imposed stress state but also by the dis-tribution of the voids and the material size effect. Theobjective of this work is to understand the interaction ofthe inter-void spacing (or ligaments) and the resultantgradient induced material size effect on void coalescencefor a range of imposed stress states. To this end, threedimensional finite element calculations of unit cell mod-els with a discrete void embedded in a strain gradient en-hanced material matrix are performed. The calculationsare carried out for a range of initial inter-void ligamentsizes and imposed stress states characterised by fixed val-ues of the stress triaxiality and the Lode parameter. Ourresults show that in the absence of strain gradient ef-fects on the material response, decreasing the inter-voidligament size results in an increase in the propensityfor void coalescence. However, in a strain gradient en-hanced material matrix, the strain gradients harden thematerial in the inter-void ligament and decrease the ef-fect of inter-void ligament size on the propensity for voidcoalescence.
In porous metals, void coalescence often drives theonset of the macroscopic flow localisation that marksthe end of uniform deformation and acts as a precursorto failure, as well as the initiation and propagation ofductile cracks [1, 2, 3]. Previous studies suggest that forconventional plasticity theory, where no material lengthscale enters the constitutive law (absence of stress/straingradient induced size effect), a decrease in the inter-void spacing promotes void coalescence [4,5] and resultsin the collapse of the yield surface [6, 7]. While for afixed inter-void spacing, it is well established that theimposed stress state has a pronounced effect on the on-set of void coalescence in conventional plasticity theory.For example, it has been shown that an increase in theimposed stress triaxiality (a ratio of the first to secondstress invariant) promotes void growth and early onsetof void coalescence [8,9,10,11]. Void coalescence is sim-ply the event where the plastic flow localises within theinter-void ligaments and successively links the neighbor-ing voids [9]. The plastic flow localisation within theinter-void ligament, however, will induce plastic straingradients that in turn may affect the strengthening andhardening of the material. This raises a fundamental a r X i v : . [ phy s i c s . a pp - ph ] N ov PROBLEM FORMULATION AND MODELLING APPROACH −
1, whereas above the thresh-old value the critical stress is smallest for a Lode param-eter value of 0. For a void in a strain gradient enhancedmaterial matrix, the value of the critical stress for voidcoalescence increases with increasing length parameteri.e. increasing gradient effect. This effect of the lengthparameter on the critical stress magnitude is found toincrease with increasing imposed stress triaxiality anddecreasing inter-void ligament size. This is because athigher stress triaxiality values and for smaller inter-voidligament sizes, there is an increase in the propensity forplastic flow localisation that introduces strong plasticstrain gradients and in turn hardens the ligament. Thismechanism leads to a decrease in the dependence of crit-ical stress on the inter-void ligament size with increasinglength parameter. The gradient induced strengtheningalso tends to homogenize the deformation in the unitcell thus decreasing the effect of the Lode parameter.The structure of the manuscript is as follows.Section 2 frames the study and presents the numer-ical method. The unit cell geometries considered,the method utilized to impose proportional loadingthroughout the deformation history, and the strain gra-dient plasticity material model are also presented in Sec-tion 2. The numerical results are presented and dis-cussed in Section 3. Finally, the key results and conclu-sions of this work are summarized in Section 4.
This work considers a limit load-type analysis todetermine the critical stress level at which a given mi-crostructure configuration loses load carrying capacity.Hence, an elastic-perfectly plastic material model is em-ployed. The configuration of the unit cell and the sim-ulation setup is described in Section 2.1, while the ap-proach to prescribe a constant value of stress triaxialityand Lode parameter is outlined in Section 2.2.
Three dimensional finite element calculations arecarried out to model the response of an array of spher-ical voids with initial radius r , Fig. 1. The unit cellhas edge lengths 2 a i along the three coordinate axes, x i ( i = 1 , , l i = 2 a i − r . Symmetry about three planes perpen-dicular to the coordinate axes implies that only 1 / f = 0 .
01, where f = (4 / πr ) / (8 a a a ).The initial void radius, r , is kept constant, while thecell dimensions are varied to achieve various initial inter- PROBLEM FORMULATION AND MODELLING APPROACH x - and x -plane. The distribution alongthe x -direction is not shown for simplicity.void spacings as in ref. [5]. The geometric parametersfor the different cell dimensions are given in Table 1.For all unit cells, a /r = a /r . Finite element meshesfor four unit cell configurations are shown in Fig. 2. Themodelling setup does not account for softening due tovoid evolution since a small strain formulation is used.It is assumed that the unit cell represents the materialcondition immediately before failure, neglecting the de-formation history leading to this state. Hence, modelpredictions for the loss of load carrying capacity signalthe onset of localisation. The critical equivalent stressat the onset of localisation is recorded and reported inthe results section.Table 1: Geometric parameters for the various unit cellsconsidered for f = 0 .
01. Based on [5]. a /r = a /r a /r l /r = l /r l /r .
06 1 .
43 5 .
05 0 . .
55 1 .
70 4 .
55 0 . .
21 1 .
94 4 .
21 0 . .
97 2 .
12 3 .
97 1 . .
58 2 .
50 3 .
58 1 . .
18 3 .
00 3 .
18 2 . .
75 3 .
75 2 .
75 2 . The unit cells are subject to prescribed displace-ments and the boundary conditions applied to the facesof the cell are (a) (b)(c) (d)Fig. 2: Finite element meshes showing 1/8 of the unitcell with an initially spherical void of radius r in thecentre giving an initial void volume fraction of f = 0 . l / r = l / r = 5 . l / r = 0 .
43, (b) l / r = l / r =4 . l / r = 0 .
95, (c) l / r = l / r = 3 . l / r = 1 . l / r = l / r = 2 . l / r = 2 .
75. The number of ele-ments ranges from 1896 for (a) to 2512 for (d). u ( a , x , x ) = U ( t ) , T ( a , x , x ) = T ( a , x , x ) = 0 u ( x , a , x ) = U ( t ) , T ( x , a , x ) = T ( x , a , x ) = 0 u ( x , x , a ) = U ( t ) , T ( x , x , a ) = T ( x , x , a ) = 0(1)The applied symmetry boundary conditions are u (0 , x , x ) = 0 , T (0 , x , x ) = T (0 , x , x ) = 0 u ( x , , x ) = 0 , T ( x , , x ) = T ( x , , x ) = 0 u ( x , x ,
0) = 0 , T ( x , x ,
0) = T ( x , x ,
0) = 0 (2)In Eq. (1), U ( t ) is prescribed and the time historyof the displacements U ( t ) and U ( t ) are determinedsuch that a prescribed stress state is maintained. Theloading direction is fixed in stress space by enforcingconstant ratios between the normal stress componentsthroughout the deformation history such thatΣ = ρ Σ , Σ = ρ Σ , (3)where ρ and ρ are constants. The overall stress com-ponents Σ ij are found by volume averaging over all el- PROBLEM FORMULATION AND MODELLING APPROACH ij = R V σ ij d V /V , where V is theunit cell volume.The overall effective stress, Σ e , and the overall hy-drostatic stress, Σ h , are given byΣ e = 1 √ q (Σ − Σ ) + (Σ − Σ ) + (Σ − Σ ) , Σ h = 13 (Σ + Σ + Σ ) , which in terms of the relative stress ratios becomeΣ e = Σ √ q (1 − ρ ) + ( ρ − ρ ) + ( ρ − , (4)Σ h = Σ
13 (1 + ρ + ρ ) . (5)The stress triaxiality, T , and the Lode parameter, L ,are given by T = Σ h Σ e = √
23 1 + ρ + ρ p (1 − ρ ) + ( ρ − ρ ) + ( ρ − (6)and L = 2Σ − Σ − Σ Σ − Σ = 2 ρ − − ρ − ρ . (7)The overall effective strain, E e , is given by E e = √ q ( E − E ) + ( E − E ) + ( E − E ) (8)where the strain components, E ij , are found in a wayanalogous to the stress components. The macroscopic normal stress components varythroughout the deformation, but the stress ratios aremaintained in each increment of the simulation accord-ing to Eq. (3). This is achieved by creating multipoint constraints through the user subroutine MPCin ABAQUS, which enables enforcing relationships be-tween degrees of freedom in one or more nodes.Additional degrees of freedom are added to imposeboundary conditions on all sides of the model whileprescribing the stress ratios. Three dummy nodes, N i , are created outside of the mesh and connectedto one connector node, M , in the mesh as shown inFig. 3. This connection is made through spring ele-ments (SPRING2 elements from the ABAQUS elementlibrary). The displacement in the x -direction is then prescribed at the N -dummy node, while the displace-ments (in the x - and x -directions) corresponding tothe desired stress triaxiality and Lode parameter are cal-culated and applied to the N - and N -dummy nodes.The displacement of the connector node, M , is coupledto the displacement of the nodes located at ( a , x , x ),( x , a , x ) and ( x , x , a ) in the direction of the respec-tive face normals. In this way, the displacement of thedummy nodes, N i , is linked to the unit cell.Fig. 3: The spring elements for the multiple point con-straints connected to one connector node, M , in thefinite element mesh.The displacement of the dummy nodes, N i , is re-lated to the forces, F i , at the faces of the unit cellthrough F i = k i ( u N i i − u Mi ) with i = 1 , , , (9)and k i being the spring element constants given by k i = E ( A i / a i ) × − , where the factor of 10 − is introducedto stabilise the numerical solution, following Ref. [34].The forces, F i , are the resultant of all traction acrossthe corresponding surface and relates to the macroscopicstresses throughΣ = F A , A = a a Σ = F A , A = a a Σ = F A , A = a a (10)where A i is the area over which the forces act. Com-bining Eqs. (3), (9), and (10), gives the dummy nodedisplacements PROBLEM FORMULATION AND MODELLING APPROACH ρ = Σ Σ = const. ⇒ u N = u M + ρ A A k k ( u N − u M ) ρ = Σ Σ = const. ⇒ u N = u M + ρ A A k k ( u N − u M ) , (11)where ρ and ρ are input values for the stress ratio, u N j i is the displacement of dummy node j in the direc-tion of x i , u Mi is the displacement in x i -direction of theconnector node M , A i are areas from Eq. (10), and k i are the spring element constants. Another relevant pro-cedure for imposing multiple point constraints withoutspring elements can be found in [35].The calculations are carried out for three valuesof Lode parameter, L = − ,
0, and 1. The Lode pa-rameter values L = − > Σ = Σ ) and L = 1(Σ = Σ > Σ ) correspond to overall axisymmetricstress states, while L = 0 (Σ > Σ > Σ ) correspondto an overall state of shear plus hydrostatic stress. Foreach value of Lode parameter, three triaxialities are con-sidered, T = 1 ,
2, and 3. The values for ρ and ρ toachieve these stress states are given in Table 2.Table 2: Input parameters determining the prescribedstress state.L T ρ ρ -1 ρ − ρ ) ρ ρ − ρ ) ρ √ − ρ ) 1+ ρ √ T − √ T +1 ρ − ρ ) T − T +1 The gradient enhanced constitutive model employedis based on the visco-plastic strain gradient plasticitytheory proposed by Gudmundson [23] in the contextof the mathematical formulation in terms of minimumprinciples proposed by Fleck and Willis [24]. For thedissipative version considered, the theory accounts forinternal elastic energy storage due to elastic strain anddissipation due to the plastic strain rate, ˙ ε pij , and itsspatial gradient, ˙ ε pij,k . Contributions from plastic straingradients to free energy is ignored. The Principle of Vir-tual Work (PVW) in Cartesian components is expressedby Z V (cid:16) σ ij δ ˙ ε ij + ( q ij − s ij ) δ ˙ ε pij + τ ijk δ ˙ ε pij,k (cid:17) d V = Z S (cid:16) T i δ ˙ u i + t ij δ ˙ ε pij (cid:17) d S (12)where σ ij and s ij = σ ij − δ ij σ kk are the Cauchy stresstensor and the stress deviator, respectively. The micro-stress, q ij , is work conjugate to the plastic strain rate,˙ ε pij , and τ ijk is a higher order stress, work conjugate tothe plastic strain rate gradient, ˙ ε pij,k . The right handside of the PVW includes the conventional traction, T i = σ ij n j work conjugate to the boundary displacementrate, ˙ u i , and the higher order traction, t ij = τ ijk n k ,work conjugate to the plastic strain rate, ˙ ε pij . Here,the outward unit normal to the surface S is n i . Balancelaws for the stress quantities are given by σ ij,j = 0 and q ij − s ij − τ ijk,k = 0 (13)where, the first set of equations is the conventional equi-librium equations in the absence of body forces, andthe second set is the higher order equilibrium equations.The higher order boundary conditions are imposed suchthat the void surface is higher order traction free, whilesymmetry conditions are imposed at the exterior of thecell through ε = 0. The rate-dependent visco-plastic formulation em-ploys a potential to account for plastic dissipation asfollows Φ (cid:2) ˙ E p , E p (cid:3) = Z ˙ E p σ c h ˙ E p , E p i d ˙ E p (14) NUMERICAL RESULTS AND DISCUSSION σ c is the gradient enhanced effective stress,related to the current matrix flow stress through σ c = σ F [ E p ] (cid:16) ˙ E p ˙ ε (cid:17) m , with ˙ ε denoting the reference strainrate, and m denoting the rate-sensitivity exponent. Thematerial in this work does not undergo strain harden-ing, making σ F independent of ˙ E p and equal to thematerial yield stress Σ . The viscoplastic law is imple-mented following the algorithm presented in Ref. [36]to efficiently approach the rate-independent limit. Agradient enhanced effective plastic strain rate is givenby (cid:0) ˙ E p (cid:1) = 23 ˙ ε pij ˙ ε pij + L D ˙ ε pij,k ˙ ε pij,k (15)and the associated work conjugate gradient enhancedeffective stress by σ c = 32 q ij q ij + 1 L D τ ijk τ ijk . (16)Here, L D is a dissipative constitutive length parameterthat enters for dimensional consistency. The superscript D refers to dissipative quantities, and the dissipativestress quantities are given by q Dij = 23 σ c ˙ ε pij ˙ E p , τ Dijk = L D σ c ˙ ε pij,k ˙ E p . (17)The dissipative length parameter controls thestrengthening size effect with an increase in the dissi-pative length parameter giving an increase in the ap-parent yield stress in the presence of strain gradients,see [37, 38]. This work is a limit load analysis, which,by definition, is done to determine the overall yield cri-terion for a given, specific configuration. Limit loadanalyses normally idealise materials as perfectly plastic.To avoid strain hardening from the energetic gradientcontributions, the energetic length parameter, L E , hasbeen set to zero in this work, and, consequently, thecorresponding energetic quantities are omitted. Throughout, the following material parameters areused; Σ /E = 0 . ν = 0 . m = 0 .
01, where Σ isthe yield stress, E is Young’s modulus, ν is the Poissonratio, and m is the strain rate sensitivity exponent. Thevalue of m is considered sufficiently small for the resultsto approximate a rate-independent material response.The influence of the Lode parameter, L , the stress triax-iality, T , and the normalised length parameter, L D /r ,is studied. The effect of the inter-void ligament size isdiscussed in combination with the other parameters, L , T , and L D /r . Figure 4 presents the equivalent stress-strain curvesfor two distinct Lode parameters, L = − T = 3, and a fixed inter-voidligament size of l /r = 1 .
5. The equivalent stress-straincurves are depicted for three length parameters, being, L D /r = 0 . , .
5, and 1 as well as for the conventionallimit where L D /r = 0.The material response shows a clear effect of plasticstrain gradients, such that the larger the length param-eter, the higher the equivalent stress level. This meansthat an increase in the stress level is obtained whendown-scaling the microstructure and, thus, yielding ofthe material is delayed due to increasing strain gradientstrengthening. The critical equivalent stress, Σ ce / Σ ,signaling localisation (and coalescence) is taken to be atthe plateau of the equivalent stress-strain curve. Severalways exists to establish a coalescence criteria based oneither critical stress or strain. The method employed inthis work is inspired by the work of [39]. The criticalstresses have been extracted from the end of the equiva-lent stress-strain curves (as shown in Figures 4a and b).Taking the example of Fig. 4a, the conventional mate-rial ( L D /r = 0), the difference between the equivalentstress at E e = 0 .
02 and 0 .
01 is less than 0 . The conventional limit, L D /r = 0, is considered toset the scene for the study of material size effects. Thefocus here is the effect of inter-void ligament size onthe critical stress at localisation under various loadingconditions.First, three values of the Lode parameter are consid-ered, L = − ,
0, and 1, for a fixed stress triaxiality, T = 2.Figure 5 shows the critical equivalent stress as a func-tion of the inter-void ligament size. For the six smallestinter-void ligaments, the critical equivalent stress is seento increase when the inter-void ligament becomes big-ger irrespective of the value of the Lode parameter. Theincrease in the critical stress ties to localisation occur-ring more easily in small inter-void ligaments loweringthe load carrying capacity of the unit cell. As the inter-void ligament size increases, the l -ligament can sustaina higher stress level before localisation, leading to anincrease in critical equivalent stress. Also, for the sixsmallest inter-void ligaments, an increase in the criticalstress is found with increasing Lode parameter values.Thus, the lowest critical equivalent stress is found for L = −
1. The dependence on the Lode parameter can berationalised by considering the imposed stress state. Incomparison to the other cases, the relative stress com-ponent, ρ , is the largest when L = − l -ligament at alower overall deformation. In contrast, the ρ takes thelowest value for L = 1, resulting in delayed localisationand the highest critical equivalent stress obtained. In NUMERICAL RESULTS AND DISCUSSION L = − (a) L = 1 (b) Fig. 4: Equivalent stress-strain curve for an inter-void ligament size of l /r = 1 . L = − L = 1 and a triaxiality of T = 3.Ref. [5], void coalescence was found to occur along theligament with the smallest applied stress for L > − l -ligament will be smallest as ρ is always thelowest stress ratio. For L = −
1, coalescence occurs in thedirection of the smallest inter-void ligament size. Thiscorresponds to the l -ligament for all geometries exceptwhen l /r = 2 .
75 as this is a perfect cube, Table 1.Fig. 5: Critical equivalent stress vs. normalized inter-void ligament size for three values of the Lode parameterwith T = 2 and L D /r = 0.There is, however, a shift in the localisation pat-tern when the l -ligament becomes sufficiently wide, forexample, a drop in the coalescence stress is found for l /r = 2 .
75 for L = 0. The load carrying capacity of thematerial increases when the distribution of voids divergefrom a regular array, i.e. when l /r = 2 .
75. However,the load carrying capacity will decrease if the arrange-ment of the voids is such that the inter-void ligament size, in any direction, is too small (e.g. l /r < l /r = 2 .
75, there is no bias to-wards the l -ligament since the unit cell takes a cubicshape. The shift in the localisation is especially promi-nent for L = 0 (a state of combined hydrostatic tensionand shear) where plastic flow localises at ≈ ◦ acrossthe cubic unit cell leading to an early loss of load car-rying capacity. The shift in the localisation is demon-strated by depicting the contours of the effective plas-tic strain for two distinct unit cells ( l /r = 1 . .
75) subjected to L = 0 and T = 2 in Figs. 6 and 7.The material response remains conventional such that L D /r = 0, and the loading conditions are described by L = 0 and T = 2. For the conventional material, thesecond term of Eq. (15) is zero ( L D = 0) and the term gradient enhanced effective plastic strain refers to thetime integration of only the first term of Eq. (15). Forthe elongated unit cell ( l /r = 1 . l , whereas locali-sation is seen to occur along two corners of the cubicunit cell ( l /r = 2 . ≈ ◦ i.e. across the diagonal.Figure 6 shows the contour of equivalent plastic strainacross the faces of the cell, while Fig. 7(a) and (b) showthe contour of the effective plastic strain in the diago-nal cross-section of both unit cells at an overall effectivestrain of E e = 0 .
03. By comparing the two contours itis seen that plastic flow is observed across the entirecross-section indicating localisation at ≈ ◦ for the cu-bic model, l /r = 2 .
75. In contrast, the plastic flow isconstricted for l /r = 1 . L D /r = 0, is shown as a function of the inter-void lig-ament size for T = 1 , , and 3 for a fixed value of theLode parameter, L = −
1. In the conventional limit, ahigh level of stress triaxiality yields low critical stressfor all ligament sizes considered. The reason being thata high stress triaxiality corresponds to higher relative
NUMERICAL RESULTS AND DISCUSSION L =0, T = 2, L D /r = 0, for l /r = 1 . l /r = 2 .
75 to the right at a macroscopic effective strainof E e = 0 .
03. For l /r = 1 .
5, localisation is favouredin the smallest ligament, l . For the cubic unit cell,however, there is no bias towards any of the ligamentsand deformation localises along ≈ ◦ , i.e. across thediagonal.(a)(b)Fig. 7: Distribution of effective plastic strain along acut from corner to corner for L = 0, T = 2, L D /r = 0at E e = 0 .
03 for two geometries: (a) l /r = 1 . l /r = 2 .
75. In (a) plasticity has not localized along45 ◦ and for this geometry localisation is favoured in thesmaller ligament, while (b) shows that a band, indicatedby the dotted line, has formed at a 45 o angle to the mainloading axis (the x -axis), thus lowering the critical ef-fective stress.stress components, ρ and ρ . Figure 8 shows littleeffect of ligament size on the critical equivalent stressfor the low value of triaxiality. For T = 1, the rela-tive stress transverse to the main loading direction isinsufficient to invoke localisation in the inter-void liga-ment and the effect of the ligament size will be limited.The cell instead undergoes macroscopic localisation and,consequently, does not exhibit a profound dependenceon the inter-void ligament size. This is in line with re-sults presented in Ref. [1], where T = 1 has been foundto be the limit below which the onset of macroscopic localisation is essentially simultaneous with void coales-cence. The results for T = 2 and T = 3 in Fig. 8 showthat the critical equivalent stress is dependent on inter-void ligament size.A small drop in the critical equivalent stress is seento occur for the largest ligament for all values of triax-iality in Fig. 8. The effect is most prominent for thehighest triaxiality, T = 3. Figure 9 shows the contourgradient enhanced effective plastic strain of the cubiccell ( l /r = 2 .
75) at an effective stress of E e = 0 .
08. Ata sufficiently large strain, plasticity is seen to initiate atthe corner opposite to the void. Due to the symmetry ofboth the loading condition (Σ = Σ for L = −
1) and theunit cell, bands of plastic deformation are observed tostretch across the x − x and x − x faces, ultimatelylowering the coalescence stress giving the drop as seenin Fig. 8.Fig. 8: Critical equivalent stress vs. normalized inter-void ligament size for three values of the stress triaxial-ities with L = − L D /r = 0. The effect of gradient strengthening in the matrixmaterial is introduced through the length parameter L D (see Section 2.3). One can imagine down-scaling the mi-crostructure when increasing the value of L D /r . Threevalues of the length parameter, L D /r = 0 . , .
5, and1, are considered in the following for all combinationsof the Lode parameter, L = − , ,
1, and stress triaxi-ality, T = 1 , ,
3. Figure 10 shows the critical effectivestress, Σ ce / Σ , as a function of the inter-void ligamentsize, l /r , for all combinations. The results obtainedfor a conventional material, L D /r = 0 are presented asa reference (see Section 3.2).The general observation is that the critical stress atlocalisation increases with the magnitude of the lengthparameter (down-scaling the microstructure). However, NUMERICAL RESULTS AND DISCUSSION l /r = 2 .
75) and L D /r = 0 at an overall equivalentstrain of E e = 0 .
08. The loading conditions appliedto give an axisymmetric stress state with L = − T = 2. Note the rotated coordinate system to show thesymmetry of the plastic flow given by the cubic unit celland ρ = ρ for L = − E p , (see Eq. (15)) for a ligament size of l /r = 1 . L = − T = 3. The contours are extracted atan overall equivalent strain of E e = 0 .
02. Figure 11(a)displays the conventional material response where lo-calisation occurs in the l -ligament. At the same levelof the overall deformation, a significantly lower effec-tive plastic strain is observed in Fig. 11(b) and (c)when increasing the length parameter. For L D /r = 0 . l -ligament, but far less than in the conventional case,while the plasticity has barely initiated at this level ofthe deformation for L D /r = 0 . E p / ˙ E e , is shown for the unitcell with the smallest inter-void ligament, l /r = 0 . L = 1 and T = 3 for theconventional material with L D /r = 0 and the mate- rial with the greatest gradient strengthening contribu-tion, L D /r = 1. Figure 12(a) shows that plastic defor-mation has developed and localised in the l -ligament,as expected for a conventional material at this loadingcondition. For the matrix surrounding the l -ligament,plasticity is reduced in favour of localisation in the l -ligament. However, for the gradient strengthened ma-terial, plasticity is not only less developed, in line withthe gradient strengthening, but also smeared out acrossthe unit cell, see Fig. 12(b). Localisation is to a littleextent observed in the l -ligament, but overall the en-tire cell experiences plasticity. This is indicative of achange in deformation mechanism along the lines of theone observed in Ref. [1] for a stress triaxiality of 1. How-ever, here it is seen with an increasing length parameter.As L D /r increases, the cell is more likely to undergosimultaneous macroscopic localisation and void coales-cence in contrast to a conventional material where thecell predominantly undergoes void coalescence for thesame loading conditions.The combined effect of the stress triaxiality (for afixed Lode parameter) and the length parameter is vi-sualised by the rows in Fig. 10, while the combined ef-fect of the Lode parameter (for a fixed stress triaxial-ity) and length parameter is visualised by the columnsin Fig. 10. Qualitatively, for a fixed stress triaxialityvalue, the length parameter has a nearly identical im-pact for all values of the Lode parameter; the criticalstress increases with increasing length parameter. It is,however, interesting that the drop in coalescence stressthe cubic unit cell ( l /r = 2 .
75) subject to L = 0 di-minishes with increasing length parameter for all valuesof stress triaxiality, Figs. 10(d)-(f). This is because in-creased gradient strengthening delays the intensificationof the plastic flow and homogenizes the plastic strainfield.For the lowest stress triaxiality value, T = 1, the ef-fect of the length parameter is small. Nonetheless, theplastic strain gradients that build up around the voidgive rise to the small increase in gradient strengthen-ing. For the loading conditions giving T = 1, the on-set of localisation is significantly delayed, thus allowingthe material to withstand higher critical stress with asmaller dependence on the inter-void ligament size. Thedeformation mechanism prevailing at this low value oftriaxiality, where macroscopic localisation and void coa-lescence occur simultaneously [1], implies that the gradi-ents surrounding the void will not influence the criticalequivalent stress to a great extent, as the deformationtakes place in the entire unit cell.For a higher value of stress triaxiality, T = 2, the ef-fect of the length parameter is more prominent as seenin Fig. 10, and the smaller the inter-void ligament size,the greater the effect of the length parameter is. Thisis because, at higher stress triaxiality values, the plas-tic flow tends to localise in the inter-void ligaments asthe ligaments diminish in size. The localisation induceslarge plastic strain gradients that in turn contribute to SUMMARY AND CONCLUSIONS L = − ,
0, and 1. For each Lode parameter, three values of the stress triaxialityare considered, T = 1 ,
2, and 3. Throughout, the parameters Σ /E = 0 . ν = 0 . m = 0 .
01 are used. Theinitial void volume fraction is, f = 0 .
01. The length parameter that enters through the gradient plasticity theoryis L D /r = 0 . , . L D /r = 0 and used as a reference.strengthening. The gradient induced strengthening inthe inter-void ligament then inhibits further plastic flowlocalisation and delays void coalescence. Although notshown here, for L = − T = 2, the gradient strength-ening is sufficiently large that increasing the value of L D /r from 1 to 2, has a negligible effect. For the in-termediate length parameter, L D /r = 0 .
5, the effectsof triaxiality and inter-void ligament size are still vis-ible but greatly reduced due to the smaller degree ofgradient strengthening. For the smallest value of thelength parameter, L D /r = 0 .
2, the critical equivalentstress values follow those of the conventional material,just at a higher relative level for all inter-void ligamentsize considered. For L = 0 and L = 1, for T = 2, the sameeffect is seen. The most pronounced effect of the lengthparameter is seen for L = − T = 3 and l /r = 0 .
43 asthis configuration has the lowest critical effective stress for the conventional material, but shows the same crit-ical stress for L D /r = 1 as in the remaining results. The interaction of the inter-void ligament size andthe gradient induced material size effect on void coales-cence is investigated for a range of imposed stress states,here characterised by fixed values of the stress triaxial-ity and the Lode parameter. To this end, three dimen-sional finite element unit cell calculations for a singleinitially spherical void embedded in strain gradient en-hanced material matrix are carried out. A conventionalmaterial matrix (absence of gradient induced strength-ening effects) is considered as reference. Increasing thelength parameter, and thereby the gradient effect, isequivalent to down-scaling the microstructure. All mi-
EFERENCES L = − T = 3, l /r = 1 . L D /r = 0, (b) L D /r = 0 . L D /r = 0 . E e = 0 .
02. The effective stress, Σ e , for the same con-figuration is shown in the cells at the bottom, also herewith (d) the conventional material, (e) L D /r = 0 . L D /r = 0 . l /r = 0 .
43 with loading condi-tions described by L = 1 and T = 3. The conventionalmaterial with L D /r = 0 is shown in a), while b) showsa gradient enriched material with L D /r = 1.crostructures considered in this work contain voids thatare below a critical flaw size [40]. Thus, plasticity theorywill reign the material response as the voids are consid-ered too small to be treated as cracks.The results for the conventional material show thatthe critical coalescence stress increases when increasingthe inter-void ligament size. The effect of the inter-voidligament size is, however, dependent on the imposedstress triaxiality, such that the effect of the inter-voidligament size increases with increasing stress triaxiality.However, above a certain threshold for the inter-voidligament size, the results show a slight decrease in thecritical stress. This drop has to do with a transitionfrom plastic flow localisation within the smallest inter- void ligament to plastic flow localization at ≈ ◦ to themain loading axis. The transition in the plastic flow lo-calisation pattern is found to be particularly pronouncedfor a Lode parameter of L = 0. However, irrespective ofthe Lode parameter value, the transition occurs as theunit cells approach a cubic geometry.For a void embedded in a strain gradient enhancedmaterial matrix, the value of the critical coalescencestress increases with increasing length parameter i.e. in-creasing the gradient strengthening effect. The effect ofthe length parameter is found to intensify with increas-ing imposed stress triaxiality and decreasing inter-voidligament size. This is due to a propensity for plasticflow localisation in the inter-void ligament when the lig-ament is small and the stress triaxiality high. Plasticflow localisation introduces large plastic strain gradientswhich in turn strengthens the ligament and delays fur-ther localisation of plastic flow. The strengthening fromplastic strain gradients also leads to a weakened depen-dency in the critical coalescence stress on the inter-voidligament size. Finally, the results show that there existsa natural upper bound where the gradient strengthen-ing is so severe that the entire matrix material yields.For very large values of the length parameter, the effectof the imposed stress state and the inter-void ligamentsize vanish, and the critical equivalent stress is identicalfor all combinations of the unit cell geometry and theloading conditions considered. Acknowledgements
This research was financially supported by the DanishCouncil for Independent Research through the researchproject “Advanced Damage Models with InTrinsic SizeEffects” (Grant no: DFF-7017-00121). The comput-ing resources provided by the high performance researchcomputing center at Texas A&M University are grate-fully acknowledged.
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