THE INFLUENCE OF DISLOCATION DENSITY ON THE BEHAVIOUR OF CRYSTALLINE MATERIALS

Transcription

1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 7, Number 4/6, THE INFLUENCE OF DISLOCATION DENSITY ON THE BEHAVIOUR OF CRYSTALLINE MATERIALS Raisa PASCAN Politehnica University of Bucharest, Faculty of Alied Sciences, 33 Slaiul Indeendentei, 64 Bucharest, Romania Abstract An elasto-lastic constitutive model is roosed to describe the behaviour of crystalline materials with microstructural defects like dislocations The defects are defined in terms of the incomatible lastic distortion The model involves the free energy assumed to be deendent on the elastic strain and torsion of the Bilby connection The non-local diffusion-like evolution equation describing the lastic distortion is derived In the case of small distortions, the edge dislocations are defined and the evolution equation for the lastic distortion tensor is derived starting from the finite distortion case The initial and boundary value roblem concerning the tensile test of a nonrectangular sheet is solved numerically in the hyothesis that, at the initial moment, a single area of dislocations describes the heterogeneity of defects Key words: Burger vector, dislocation density, diffusion-like evolution for lastic distortion, variational equality, FEM and udate algorithm INTRODUCTION In this aer, we resent a model within the constitutive framework roosed in [], by considering dislocations only as defects, without disclinations The results rovided here aear as articular cases of those obtained in the above mentioned aer The measures of defects are defined in terms of the incomatible lastic distortion, F The defect densities characterize the Burgers vectors defined within the finite deformation framework, and are reduced to curl H in the case of small lastic distortion Comaring to [8] where the scalar densities of dislocations in sli systems have been considered, in the resent model we adot tensorial densities of dislocations Also, in [8, 9] non-local diffusion-tye equations for scalar densities of dislocations were reresented In the numerical examle resented in [7], at the initial moment, a non-homogeneous distribution of the scalar densities of dislocations was assumed, only the local tye of the evolution equations being considered In Section we recall the definition of the Burgers vector, and tensorial definitions for the dislocation densities In Section 3 the rincile of the free energy imbalance is formulated like in [5] (following the idea of Gurtin [4, ] The viscolastic-like equation for micro forces and a new evolution equation for the lastic distortion were derived, with resect to the reference configuration within the large deformation formalism In Section 4 we restrict to the small elasto-lastic distortions formalism and the evolution equation is also rovided In section 5, the corresonding variational equalities for the incremental equilibrium equation, and the evolution equation for the lastic distortion were formulated In Section 6 we consider a tensile roblem in a non-rectangular sheet when dislocations describe the initial heterogeneity of the defects The initial and boundary value roblem is formulated Notations For a second-order tensor A Lin and a third-order tensor Aij Γ Lin ( V,Lin) { N: V Lin, linear}, we denote: ( A ) = the gradient comonents of the field ijk k X

2 The influence of dislocation density on the behaviour of crystalline materials 359 A ; SkwΓ is defined by ((Skw Γ ) uv ) = ( Γuv ) ( Γ vu ), uv, V ; curla is defined by Ak (curl A) i = ε ijk, where ε j ijk denote Ricci ermutation tensor comonents; The gradient with resect to X the configuration with torsion K is given by ( Au ) = ( A)( A) u, u F V ; The second order velocity χ ijk k j x x i gradient is defined by ( ) ( ) ( [, ] ) ( ( )) K v L, with = χ Γ F F u v= Γ Fu Fv, uv, V, where F, symmetric and skew-symmetric arts of A L v; [ F, F ] Γ is defined by S F are second order tensors; { }, { } A A are the a MEASURES OF DISLOCATIONS The behaviour of elasto-lastic crystalline materials with defects is described in terms of three configurations: the reference configuration k of the body B, k( B) E, χ (, t) the deformed configuration at time t, for any motion of the body B, χ : B E and the so-called current local relaxed configuration, or lastically deformed configurations K The deformation gradient F= χ( X, t) is multilicatively decomosed into the elastic and lastic e e distortions, denoted by F, F, namely F= F F, at every oint of the body, using the hysical arguments mentioned in [] The elastic and lastic distortions F e and F are invertible tensors which cannot be exressed through the gradient of certain vector fields Let Α be a surface with the normal N bounded by C a closed curve in the reference configuration Following [4] the Burgers vector associated with the circuit C is defined by b = dx = F d X = (curl F ) Nd A= (curl F )( F ) n d A T K K K K detf CK C A AK () The Noll's dislocation density tensor α K with resect to the configuration with torsion is exressed by T αk : = (curl F )( F ), and was introduced by [7] By a ull-back rocedure we define the detf dislocation density tensor α with resect to the reference configuration α : = ( F ) curl F We refer to the lastic distortion as an incomatible field The lastic distortion has a non-vanishing curl, ie curl( F ) in a material neighbourhood of the considered material oint, which ensures the existence of a non-vanishing Burgers vector defined by the relation () We define the lastic Bilby connection [3] in terms of the gradient of lastic distortion with resect to the reference configuration, k, by ( ) A = ( F ) F Consequently, the tensorial measure of dislocation α also means that ( ) ( ) ( ) ( ) α( u v) = ( Au) v ( A v) u= (Skw A) u v, u, v V 3 FREE ENERGY IMBALANCE AND MICRO BALANCE EQUATIONS ELASTO-PLASTIC MODELS WITH DISLOCATIONS IN THE REFERENCE CONFIGURATION 3 Free energy imbalance We assume the existence of the free energy density function and the definition of the internal ower ( P ), which includes the work done by forces conjugated with the aroriate rate of second order elastic int K

3 36 Raisa Pascan 3 and lastic deformations The effect of dislocations is involved in the model via the gradient of the lastic distortion AXIOM The elasto-lastic behaviour of the material is restricted to satisfy the free energy imbalance formulated in K and written for any virtual (isothermal) rocess, ie ( P int ) ψ K K In the following,, are the mass densities in the initial, relaxed and actual configurations resectively, and they are related by J =, J =, where J = detf, J = detf AXIOM The internal ower in K configuration is ostulated to be given by P Y ( ) () S e e e e ( int ) K = { T} L + L + μ KL + μk ( F ) ( χl)[ F, F ] KL Here the symmetric art { T } S of the Cauchy stress tensor T is conjugate with the rate of elastic distortion, L= FF e e e, L = F ( F ), L = F ( F ) The macro stress momentum μ K, written in the last term of (), is ( ) work conjugate with a measure of the gradient of the rate of elastic distortion The micro forces Υ, μ are associated with the lastic behaviour Macro balance equations for the non-symmetric Cauchy stress, T, and macro momentum μ satisfy in the actual configuration the following balance equation S a div ({ T} {div μ} ) + b = (3) We recall that the macro stress momenta, μ and μ K, in the actual and anholonomic configuration are μ e T μ T T related by ( ) K e = F [( F ),( F ) ] For micro balance equations related to the lastic behaviour, we refer to [4,6] PROPOSITION The micro balance equation for micro forces associated with the lastic mechanism is written in the reference configuration as J Υ J K in B on B (4) T T = div( ( μ μ )( F ) ) + B, μ ( F ) N= M In the following we ostulate that the free energy density with resect to the reference configuration deends on the elastic strain and defects in the form ψ = E ( C C ) ( C C ) + β S S, where E is the 4 T T elastic stiffness matrix, C= F F, C = ( F ) F and S is the skew-symmetric art of the connection ( ) ( ) ( Suv ) (Skw A ) uv Here β, is a real arameter The consequences induced by the rincile of the free energy imbalance are the following: PROPOSITION The constitutive equations are reduced to the elastic constitutive one: E (5) S T { T} = F ( C C ) F, μ =, and the energetic relationshis which exress the lastic micro momentum as The evolution equations for the lastic distortion are given by: μ = β S (6)

4 4 The influence of dislocation density on the behaviour of crystalline materials 36 Σ + C E ( C C ) = ξ l, where l = ( F ) F (7) Here Mandel stress tensor, is introduced with resect to the reference configuration by T T ( ) ( ) Σ = F Υ F, while the micro stress momenta with resect to the reference configuration are exressed by μ F μ F F T = ( ) [( ),( ) ] 4 MODEL WITH SMALL ELASTIC AND PLASTIC DISTORTIONS Next, we consider the model of small distortions In the case of small elastic and lastic distortions, the linearized exressions derived from the finite deformation fields are given by ( ) S F I+ H, H= u, ε = { H}, F = I+ H, A H, S ε = { H }, C C = ( ε ε ), C = I+ ε, (8) where u is the dislacement vector The torsion tensor S is exressed by S = Skw H The energetic constitutive equations are given by T = E ( ε ε ), μ = β Skw H (9) The micro forces are characterized by Σ = div( μ ) = β curl(curl H ) as a consequence of its own balance equation (4) Under the hyothesis of small distortions, the elasto-lastic roblem is characterized, in terms of the dislacement vector u and lastic deformation tensor H, by E ε ε = ε= u+ u ε = H () T S div( ( )), ( ), { } ( ) The incremental equilibrium equation is obtain by deriving the above relation with resect to time and takes the form T S div E ( v+ ( v) ) { H } = () The evolution equation for the lastic distortion H is obtained from (7) and is given by ξ H = β curl(curl H ) + T () In order to obtain a boundary value roblem we attach the boundary condition for the incremental equilibrium equation and for the evolution equation for the lastic distortion Let Ω be the domain occuied by the body B at the moment t and Γ= Ω be the boundary of Ω We assume the following boundary conditions for the incremental equilibrium equation: Tn = t on Γ, v = v on Γ, (3) * where Γ Γ =Γ and Γ Γ =, and for the evolution equation for the lastic distortion: α( εn) = (curl H )( ε n) = h on Γ, (4)

5 36 Raisa Pascan 5 where n is the outward normal to the boundary of the domain Ω and ε reresent Ricci ermutation tensor 5 THE WEAK FORMULATION IN THE CASE OF SMALL DISTORTIONS The corresonding variational equality for the incremental equilibrium equation is given by: Ω ( ( ) T ) { } S S E v+ v H { } d d A =, ad w x tw w V (5) Γ The unknown v V { : 3 * ad = v Ω R v= v on Γ} and V = 3 ad { w Ω R w = Γ } : on The weak form of the evolution equation for the lastic distortion can be characterized for any field G ξ H Gdx = β (curl H ) (curl G)dx+ β h Gd A + TG d x (6) Ω Ω Ω Ω time Let Let us consider ( ) 5 The discretization of the weak form for the evolution equations tn n N a artition of the time interval [ ] n Ω be the domain occuied by the body on the [, ] n n,t and t = t + t be the increment of d n n t t + interval For the discretization we aly an imlicit rocedure The time derivative of the lastic distortion is aroximate by the formulae n+ n H H H and the lastic distortion we aroximate by H H n+ With these considerations (in the dt h = hyotheses) the discretization of the weak form for the lastic distortion becomes: H H G d x β (curl H ) (curl G )d x T n G d x (7) dt n+ n ξ = n+ + Ωn Ωn Ωn 5 The discretization of the weak form for the incremental equilbrium equation tye In the same way we obtain the discretization for the variational equality (5): Ωn E S T H ( ( ) ) n+ H n S t { } d n+ tn vn + v n w x w d A = Γn dt dt (8) 6 EDGE DISLOCATIONS An edge dislocation is characterized by the comonents of the lastic distortion which generates a Burgers vector in the lane ( e, e ), ie b e b b 3 b= + e e The lastic distortion is given by: ( ) ( ) ( ) H = H e e + H + H + H ( ) ( H = H ), x, i, j {,} The e e e e e e, where ij ij ( x ) ( ) ( ) ( ) ( ) H H H H dislocation tensor takes the following form: α= curl H = ( ) e e + ( ) e e 3 3 x x x x In our examle it is assumed that the initial existing defects inside the microstructure are reduced to an area of dislocations, which is characterized by the dislocation tensor α ( x) To find the initial condition for H ( x ), ie H ( ) x, we follow the rocedure roosed by [] and [8]: ( ) curl H e = b, divh =, x Ω, H =, x Ω (9) 3 This means that the following roblems has to be satisfied by H i, j {,}, resectively ( ) ij,

6 6 The influence of dislocation density on the behaviour of crystalline materials 363 H + H = b x Ω, H =, x Ω ; H + H = b x Ω, H =, x Ω;,,,,,, H + H = b x Ω, H =, x Ω ; H + H = b x Ω, H =, x Ω,,,,,, () We consider a distribution of Burger vectors arallel to the Ox axis, ie b = be In the numerical su su ( x x ) ( x y ) simulation the function b is defined by b ( t) = bmax ex[ k( + )], ( x, x) Ω ax ay The algorithm for solving the system of the equations (), (5) and (6): The elastic roblem is resolved until the averaged value of the equivalent stress state becomes equal or larger than a critical value, ie T T d x / A( Ω) < /3σ y Assume that at time t the stress reached the yield condition We mention that ( ) Ω A Ω reresent the area of the domain Ω At this moment we solve roblems () by emloying the finite element method (FEM) The fields ( ) H ( ), ij t reresent the initial conditions for the evolution equation () For tn t We suose that at the moment t n one knows the current state of the body, namely: Hn, v n By an imlicit rocedure we find the solutions for H at the moment of time, t n + namely H n+ ; We return at the discretisation of the weak form for equilibrium equation for the rate of dislacement v and we find the solution at the moment of time t, n + namely v n+ One can udate the mesh and all the measures calculated on the revious mesh, knowing the velocity field v n+ The rocedure continues In the numerical simulations, a bidimensional domain reresented in Fig, occuied by an aluminum crystal has been considered In the tensile test, the edge x = (left side) is fixed The numerical algorithms were erformed using the FreeFem++ ackage [6] The edge x = L (right side) is moved with a constant seed v = 5 nm /μs alied along the axis Ox and is fixed along the axis Ox The time integration 5 ste (time increment) dt = μs generates an incremental elongation dε = 5 The initial ext dimensions of the sheet are: total length L = 3 nm, the maximum width l = 65 nm and the minimum int width l = 5 nm 3 The values of the material arameters for aluminum []: μ = 7 MPa, ν = 3, σ = 7MPa, E μ(- ) μ = E ν ν =, = =, = = = = μ -ν E E -ν E E E E The arameters aearing in the 5 3 evolution equations are numerically evaluated in []: ξ = 57 μsma, β = 57 nm MPa The geometrical arameters which characterize the initial values of the dislocations area are: a x = 5 nm, a y = nm, x = L/nm, y = l/nm, k = 46 In Fig the initial inhomogeneity, reresented by the dislocation tensor comonent ( curl ) y b = H is lotted at the moment when the lastic deformation is initialized (Fig a) and at the total tensile strain of % (Fig b) In the center of the sheet we observe the diffusion effect, as the defects are sreading, with an increase of the tensile strain In Fig are reresented the solution of the roblems (), ie the distribution of the lastic distortion tensor: H ( t ) and H ( t ), in the initial lastic state that corresonds to ε =% The values of the equivalent deviatoric stress in the defect region are comarable with the values which aeared in the geometric concentration area (Fig 3b) The elastic strain fields show dilatation and contraction in the defect zone (Fig 4b) and shear along the vertical axis of the defect area (Fig 4a) In Fig 3a, the equivalent lastic deformation is shown, and we remark the hardening of the material in ec the zone corresonding to the maximum value of T = T T and the softening of the material in the zone corresonding to the minimum value of the equivalent deviatoric stress 3

7 364 Raisa Pascan 7 These numerical results are similar with the case in which is considered a diole of disclinations only In the aer [] we emhasized that the Burgers vectors of the edge dislocations ( α 3, α 3 ), which are inside the disclination diole, is aroximately normal to the diole arm This result is in agreement with the exeriment [] The numerical test simulated in the resent aer shows that a zone with normal Burgers vectors to the Ox axis roduce such effect similar to that of a diole of disclinations with the axis in the Ox direction x O x (a) (b) b = H in nm : Fig Distribution of the dislocation tensor comonent ( curl ) a) in the initial lastic state that corresonds to ε =% ; b) at the total tensile strain of % 3 (a) Fig Distribution of the lastic distortion tensor: a) H ( t ) ; b) ( ) (b) H t in the initial lastic state (a) (b) Fig 3 Distribution of: a) the equivalent lastic deformation ec T H = H H ; b) of the equivalent deviatoric stress ec = T T in MPa, at the total tensile strain of % (a) (b) e Fig 4 Distribution of: a) the elastic comonent ε ; b) of the trε e, at the total tensile strain of %

8 8 The influence of dislocation density on the behaviour of crystalline materials 365 CONCLUSIONS The model describes the behaviour of crystalline materials with defects defined in terms of the incomatible lastic distortion The model was roosed within the second order elasto-lasticity develoed by Cleja-Tigoiu [4, 5, 8]The non-local diffusion like evolution equation for lastic distortion was derived to be comatible with the reduced dissiation inequality for finite and small deformation In order to validate the elasto-lastic model with edge dislocations, the initial and boundary value roblem concerning the tensile test of a non-rectangular sheet was solved numerically in the hyothesis that, at the initial moment, a single area of dislocations describes the heterogeneity of defects The numerical results clearly showed the effects induced by the initial heterogeneity ACKNOWLEDGMENTS Raisa Pascan was suorted by the strategic grant POSDRU/59/5/S/3775, Project Doctoral and Postdoctoral rograms suort for increased cometitiveness in Exact Sciences research cofinanced by the Euroean Social Found within the Sectorial Oerational Program Human Resources Develoment 7 3 REFERENCES Acharya, A, New inroads in an old subject: Plasticity, from around the atomic to the macroscoic scale, J Mech Phys Solids, 58, , Beausir, B, Fressengeas, C, Disclination densities from EBSD orientation maing, Int J Solids Struct, 5, 37 46, 3 3 Bilby, B A, Continuous distribution of dislocations, Sneddon IN, Hill R (eds), Progress in Solid Mechanics, North-Holland, Amsterdam, 96, Cleja-Tigoiu, S, Material forces in finite elasto-lasticity with continuously distributed dislocations, Int J Fracture, 47, 67 8, 7 5 Cleja-Tigoiu, S, Elasto-lastic materials with lattice defects modeled by second order deformations with non-zero curvature, Int J Fracture, 66, 6 75, 6 Cleja-Tigoiu, S, Non-local elasto-viscolastic models with dislocations in finite elasto-lasticity Part I: Constitutive framework, Math Mech Solids, 8, , 3 7 Cleja-Ţigoiu S, Paşcan R, Influence of Dislocations on the Deformability of Metallic Sheets, Key Engineering Materials, Trans Tech Publications, , 99 9, 3 8 Cleja-Tigoiu, S, Pascan, R, Sli systems and flow atterns in viscolastic metallic sheets with dislocations, Int J Plasticity, 6, 64 93, 4 9 Cleja-Ţigoiu S, Paşcan R, Non-local elasto-viscolastic models with dislocations in finite elasto-lasticity Part II: Influence of dislocation density, Math Mech Solids, 8, 4, , 3 Cleja-Tigoiu, S, Pascan, R, Tigoiu V, Elastolastic models with continuously distributed defects: dislocations and disclinations, for finite and small strains, Int J Plasticity, submitted Cleja-Tigoiu, S, Soós, E, Elastolastic models with relaxed configurations and internal state variables, Al Mech Rev, 43, 3 5, 99 Fressengeas, C, Tauin, V, Caolungo, L, An elasto-lastic theory of dislocation and disclination fields, Int J Solids Sruct 48, , 3 Gudmundson, P, A unified treatment of strain gradient lasticity, J Mech Phys Solids, 5, , 4 4 Gurtin, ME, On a framework for small-deformation viscolasticity: free energy, microforces, strain gradien, Int J Plasticity, 47 9, 3 5 Gurtin, ME, Fried E, Anand, L, The Mechanics and Thermodynamics of Continua, Cambridge Uniniversity Press, 6 Hecht, F, Auliac, S, Pironneau, O, Morice, J, Le Hyaric, A, Ohtsuka, K, FreeFem++, htt://wwwfreefem org/ff++/ft/freefem++docdf, (Accessed may, 4) 7 Noll, W, Materially uniform simle bodies with inhomogeneities, Arch Rat Mech Anal, 7, 3, Tauin, V, Caolungo, L, Fressengeas, C, Das, C, Uadhyay, M, Grain boundary modeling using an elasto-lastic theory of dislocation and disclination fields, J Mech Phys Solids, 6, , 3 Received August, 5