Elsevier Editorial System(tm) for Computer Methods in Applied Mechanics and Engineering Manuscript Draft

Elsevier Editorial System(tm) for Computer Methods in Applied Mechanics and Engineering Manuscript Draft Manuscript Number: Title: Modelling of Strain Softening Materials Based on Equivalent Damage Force Article Type: Research Paper Keywords: strain softening instability, FEM, nonlocal damage, composite materials, quasi brittle materials Corresponding Author: Professor Rade Vignjevic, Corresponding Author's Institution: Brunel University London First Author: Rade Vignjevic, Prof Order of Authors: Rade Vignjevic, Prof; Rade Vignjevic; Nenad Djordjevic, Dr; Tom DeVuyst, Dr; Simone Gemkov, Dr Abstract: The main aim of the work presented in this paper was addressing localisation problem observed in the analysis of strain softening materials using finite element methods (FEM) combined with local continuum damage mechanics (CDM) approach. Strain softening is typically observed in damaged quasi brittle materials such as fibre reinforced composites and application of the CDM approach with the classic FEM features a number of anomalies, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). These features of the classic FEM solutions have been already demonstrated in (Vignjevic, Djordjevic et al. 2014). The model proposed here is still based on the local CDM approach, but introduces an alternative definition of damage effects in the system of equilibrium equations. The constitutive equation in the model is defined in terms of effective stress, whilst the damage effects in the conservation of momentum equation are calculated as equivalent damage force (EDF), which contributes to the equilibrium on the right hand side of the momentum equation. The main advantages of this model are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening, numerical stability, which is preserved without a need for regularisation measures, and significantly reduced mesh dependency. In addition, the EDF model can be combined with existing local CDM damage evolution functions and does not violate symmetry of the stiffness tensor. The EDF model was implemented in in-house developed coupled FEM - MCM code, where explicit FEM (Liu 2004) is coupled with a stable Total- Lagrange form of SPH (Vignjevic, Reveles et al. 2006, Vignjevic, Campbell et al. 2009). Its performance is demonstrated in the analysis of a dynamic one dimensional stress wave propagation problem, which was analytically solved in (Bazant, Belytschko 1985). For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be

experimentally determined and is an input parameter independent of the discretisation density. Suggested Reviewers: John P Dear Prof, Dr, MSc Prof, Mechanics of Materials Division, Imperial College London j.dear@imperial.ac.uk Expertise in mechanics of materials computational mechanics Vassilis Kostopoulos Prof, Dr, MSc Director of the Laboratory of Applied Mechanics and Vibrations, Department of Mechanical Engineering & Aeronautics, University of Patras kostopoulos@mech.upatras.gr Expertise in computational mechanics and mechanics of materials Michele Meo Prof, Dr, MSc Prof, Material Research Centre, Dept of Mechanical Engineering, Bath University m.meo@bath.ac.uk Expertise in computational mechanics and mechanics of materials Miroslav Zivkovic Prof, Dr, MSc Prof, Depatment for Applied Mechanics, University of Kragujevac zile@kg.ac.rs Expertise in computational mechanics and mechanics of materials Mauro Zerrilli Dr, MSc Senior Scientist, IPCB Institute, CNR Research National Council of Italy mauro.zarrelli@cnr.it Expertise in mechanics of materials and material modelling George S Dulikravich Prof, Dr, MSc Prof, Department of Mechanical and Materials Engineering, Florida International University dulikrav@fiu.edu Expertise in computational mechanics and mechanics of materials Opposed Reviewers:

Cover Letter Dynamic Response Group Brunel University London, Kingston Lane, Uxbridge UB8 3PH, United Kingdom Email: v.rade@brunel.ac.uk Tel: +44 (0)7786748151 Prof. Thomas J.R. Hughes Editor of Computer Methods in Applied Mechanics and Engineering 30 October 2017 Dear Prof. Hughes, I have submitted paper titled: Modelling of Strain Softening Materials Based on Equivalent Damage Force to be considered for publication in the Computer Methods in Applied Mechanics and Engineering journal. The paper is addressing localisation problem typical for the analysis of strain softening materials using finite element method combined with local constitutive models and continuum damage mechanics. The damage effects in the conservation of momentum equation are represented as an equivalent damage force, which contributes to the right-hand side of the momentum equation. This allows for the problem to remain well posed. Numerical stability, is preserved without a need for regularisation measures. Your sincerely Prof Rade Vignjevic

Highlights (for review) Highlights Proposed an approach for treatment of damage localization in softening materials Damage effects represented by an equivalent damage force Localization damage zone size defined by a characteristic length The approach is compatible with local constitutive equations

*Manuscript Click here to download Manuscript: Equivalent Damage Force_Manuscript_5.pdf Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Modelling of Strain Softening Materials Based on Equivalent Damage Force Rade Vignjevic 1,*, Nenad Djordjevic 1, Tom De Vuyst 1, Simone Gemkow 2 1 Dynamic Response Group, Structural Integrity Theme, Brunel University London, Kingston Lane Uxbridge, UB8 3PH, United Kingdom e-mail: v.rade@brunel.ac.uk 2 Cranfield University, Cranfield, Bedfordshire MK43 0AL, United Kingdom Abstract The main aim of the work presented in this paper was addressing localisation problem observed in the analysis of strain softening materials using finite element methods (FEM) combined with local continuum damage mechanics (CDM) approach. Strain softening is typically observed in damaged quasi brittle materials such as fibre reinforced composites and application of the CDM approach with the classic FEM features a number of anomalies, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). These features of the classic FEM solutions have been already demonstrated in (Vignjevic, Djordjevic et al. 2014). The model proposed here is still based on the local CDM approach, but introduces an alternative definition of damage effects in the system of equilibrium equations. The constitutive equation in the model is defined in terms of effective stress, whilst the damage effects in the conservation of momentum equation are calculated as equivalent damage force (EDF), which contributes to the equilibrium on the right hand side of the momentum equation. The main advantages of this model are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening, numerical stability, which is preserved without a need for regularisation measures, and significantly reduced mesh dependency. In addition, the EDF model can be combined with existing local CDM damage evolution functions and does not violate symmetry of the stiffness tensor. The EDF model was implemented in in-house developed coupled FEM MCM code, where explicit FEM (Liu 2004) is coupled with a stable Total-Lagrange form of SPH (Vignjevic, Reveles et al. 2006, Vignjevic, Campbell et al. 2009). Its performance is demonstrated in the analysis of a dynamic one dimensional stress wave propagation problem, which was analytically solved in (Bazant, Belytschko 1985). For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density. Keywords strain softening instability, FEM, nonlocal damage, composite materials, quasi brittle materials Page 1 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1 Introduction Strain softening is deterioration of stress for increasing strain, which is a phenomenon typically observed at a continuum level in damaged quasi brittle materials, including fibre reinforced composites. It is primarily a consequence of brittleness and heterogeneity of the material and it has been experimentally demonstrated the strain softening in the material is distributed over a finite region, see for instance (Bazant, Belytschko et al. 1984) and references therein. An existing approach to the strain softening problem in Continuum Damage Mechanics (CDM) is to model degradation of material properties as a loss of effective load-carrying area (Kachanov 1958, Lemaitre 1985, Krajcinovic 1996), which smears micromechanical damage processes occurring in the softening zone. When the CDM model is implemented as local model in the finite element method (FEM) code, the strain softening leads to numerical instability, as the tangent stiffness (slope of the stress stain curve) loses positive definiteness and violets the material stability criterion by Hadamard (Hadamard 1903). Consequently, the underlying initial boundary value problem becomes ill-posed and the continuum solution bifurcated, leading to an infinite number of solutions. In addition, implementation of these local CDM models in the numerical codes leads to the localised deformation and pronounced sensitivity of the results to the spatial discretisation (mesh density), as already demonstrated in (Vignjevic, Djordjevic et al. 2014). Localised deformation and mesh sensitivity mean infinite increase of the local strain upon mesh refinement, which in the limiting case, result in physically unrealistic energy dissipation of a damaged zero volume element. In summary, strain softening brings in mathematical pathology, in terms of change of the type of partial differential equations (PDE), numerical pathology, in terms of mesh sensitivity, and leads to the physically meaningless results. The strain-softening instabilities have been of large interest to research in recent decades and have been investigated, among many others in (Pijaudier-Cabot, Bazant et al. 1988, Neilsen, Schreyer 1993, Sluys, de Borst 1994a), leading to a development of a number of regularisation methods, including non-local, gradient-enhanced and viscous methods. These methods are based on the introduction of a characteristic length scale into constitutive equations through higher-order spatial derivatives or viscos effects, see for instance the models developed by Dillon (Dillon Jr., Kratochvil 1970), Bazant (Bazant, Zubelewicz 1988, Bazant, Jirasek 2002), Aifantis (Aifantis 1984, Aifantis 1992), Needleman (Needleman 1988, Tvergaard, Needleman 1995, Tvergaard, Needleman 1997), Pijaudier- Cabot (Pijaudier-Cabot, Bazant 1987, Pijaudier-Cabot, Bazant et al. 1988), Sluys (Sluys, de Borst 1992, Sluys, de Borst 1994b), and de Borst (Peerlings, De Borst et al. 1998, Peerlings, Geers et al. 2001, Peerlings, De Borst et al. 2002). These regularisation methods prevent development of the material instability i.e. maintain the type of underlying governing equations, which are elliptic partial differential equations in static problems and hyperbolic in dynamic problems, which in turn lead to a well-posed initial boundary value problem. The internal length scale confines the area affected by strain-softening to a finite size providing physical and mesh-independent finite element solutions. Despite the evident success of regularisation methods in the field of strain-softening instabilities, research has been almost exclusively focused on these methods and, to date, there has been little research into solutions based on local constitutive equations. However, this might be of interest to users of strain-softening models as regularisation methods necessitate an increased understanding of the underlying strain-softening problem, definition of the characteristic length for the material of interest and make the application of regularisation methods numerically more expensive. More Page 2 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 importantly, a suitable definition of damage effects present in a continuum, combined with existing CDM damage functions, allows for more flexibility in terms of formulation and characterisation of the constitutive models. We already demonstrated that the SPH method was inherently non-local method and that the smoothing length could be linked to the material characteristic length scale in solid mechanics simulations, whilst the FEM results in presence of strain-softening were unstable and highly mesh sensitive (Vignjevic, Djordjevic et al. 2014). Consequently, the aim of the work presented in this paper was development of an alternative new approach to modelling damage in strain softening materials, which was compatible with FEM and based on the existing local constitutive equations. The model developed is called equivalent damage force (EDF) and the key feature of this approach is that contribution of damage to the conservation of momentum is calculated as a force on the right hand side of the dynamic equation of motion. The proposed EDF method maintains a well-posedness of initial boundary value problems and, therefore, does not require any regularisation measures within constitutive equations in modelling strain-softening materials. In addition, the method can be combined with any CDM local damage evolution law, yet providing the mesh independent stable solutions. This paper consists of five sections. Following the introduction to the strain softening problem and associated issues, a benchmark dynamic strain softening problem is described in Section 2 with the analytical local and nonlocal solutions available from (Bazant, Belytschko 1985). The Equivalent Damage Force model is presented in Section 3, including the derivation of principle equations and model implementation in the in house developed coupled FEM-MCM code. The model is validated against the known analytical solutions in Section 4, with the outcomes of this work summarised in Conclusion in Section 5. 2 Dynamic strain softening problem 2.1 Analytical solution Development of localised deformation is a result of the physical processes occurring in the material at microscale, including initiation, growth and interaction of cracks and voids, which finally lead to complete material failure. In this investigation, a definition of localisation proposed in (Rudnicki, Rice 1975) is used: Localization is defined as instability in the macroscopic constitutive description of inelastic deformation of the material. Damage evolution in local constitutive equations for a homogeneous material leads to a bifurcation point, where the material becomes unstable and the deformation localises within an infinitely small instability zone and becomes non-uniform. Outside this instability zone the material remains stable (Rudnicki, Rice 1975). Material is stable and stays in equilibrium when the double contraction of stress rate Page 3 of 16 ij and strain rate ij, given in (1.1) is positive. This criterion is also called general bifurcation criterion (Neilsen, Schreyer 1993), and is satisfied as long as the material behaviour is determined with a positive definite stiffness tensor.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 (1.1) ij The rate form of constitutive equations used here ensures a piecewise linear relationship between stress-rate and strain-rate, which can be expressed as a constitutive equation defined in terms of material tangent stiffness tensor ijkl : So that the inequality (1.1) reads: ij (1.2) ij ijkl kl 0 (1.3) ij ijkl kl The material becomes unstable when it reaches its bifurcation point i.e. when the condition in (1.3) is violated. Hence, the bifurcation point is defined as: 0 (1.4) ij ijkl kl Condition (1.4) is satisfied when the tangent stiffness tensor becomes singular, i.e. positive-definite anymore, which corresponds to zero stiffness tensor determinant: Page 4 of 16 ijkl is not det ijkl 0 (1.5) The initial analytical and later numerical investigation of strain-softening was carried out by investigating longitudinal wave propagation in a bar shown in Error! Reference source not found.. This problem in context of local strain-softening continua was first considered and analytically solved in (Bazant, Belytschko 1985) and has been repeatedly used by researchers in the field of strainsoftening within the CDM framework and the investigation of regularisation methods, see for instance (Bazant, Zubelewicz 1988, Graff 1991, Sluys, de Borst 1994b, Peerlings, Geers et al. 2001). The problem is one-dimensional, which simplifies interpretation of the resulting strain-softening effects and provides clarity. One dimensional stress wave propagation problem is 2L long bar, symmetrically loaded at both ends with a constant velocity v. In the original paper (Bazant, Belytschko 1985), material behaviour of the bar was determined by stress strain relationship illustrated in Error! Reference source not found.b, where the softening zone between Point P and Point F, was characterised with a negative slope and elastic unloading/reloading law. The symmetric loading of the bar generates two tensile step stress waves, which propagate towards the midsection of the bar ( x 0 ), where they are superposed at time t L / c. Superposition of the strain waves in the midsection of the bar instantaneously doubles the strain at that point, which can result in strain softening material behaviour. The time and location of strain-softening occurrence will be determined precisely using wave propagation theory, outlined below.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Location for Figure 1 The equation of motion is derived from well-established wave equation, valid for elastic and inelastic material behaviour, which is in its standard hyperbolic form for elastic behaviour given as: c 2, ux, t 2 2 u x t x 2 2 where: c is elastic speed of sound, ux, t longitudinal displacement, x longitudinal coordinate and t is time. Please note that the elastic speed of sound of isotropic material, for the uniaxial stress and uniaxial strain state propagation problem are respectively defined as: c Page 5 of 16 t (1.6) E c (1.7) E 1 12 1 With E and being Young s modulus and Poisson s ratio. Analytical solution for this stress wave propagation problem can be derived starting from for the longitudinal displacement function used in analysis of elastic longitudinal wave propagation in a semi infinite bar proposed in (Von Karman, Duwez 1950) as: (1.8) x L x L u x, t v t v t (1.9) c c Where the brackets represent positive definite expressions and L is half of the bar length. The corresponding strain x and stress x components in the loading direction are defined as (Vignjevic, Djordjevic et al. 2014): Where H is Heaviside function. u v x L x L x H t H t x c c c x E 1 12 1 Superposition of the strain waves in the midsection of the bar ( x 0 ) at response time t L / c instantaneously doubles the strain at that point, x vc, so depending on the impact loading condition, one can distinguish three possible scenarios: x (1.10) (1.11)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1) when the strain obtained in the midsection of the bar satisfies condition 2, the bar is elastically loaded during the whole loading process, and assumption of linear elasticity holds even after the waves superposition, i.e. until both waves travel the whole bar length; 2) when the strain obtained in the midsection of the bar satisfies condition 2, the Page 6 of 16 x p p x p bar is elastically loaded for the time 0 t L c ; however, at t L c, strain instantaneously enters the strain-softening regime; in that case, solution for the longitudinal displacement given in Equation (1.9) holds only for the elastic part of the response, i.e. t L c ; 3) when the strain obtained in the midsection of the bar satisfies condition x p, the bar undergoes inelastic wave propagation, which is not considered in this paper; Please note that p, which is elastic limit for the model, need to be calculated in line with the constitutive equations for the uniaxial stress and uniaxial strain states, where the latter is given in Equation (1.10). The objective of this work is analysis of the second scenario given above, where following the superposition of the tensile waves at t L c, the slope of the stress strain curve in Error! Reference source not found., becomes negative, i.e. F 0, and wave speed becomes imaginary so that the equation of motion in the softening domain becomes an elliptic PDE: c 2, ux, t 2 2 u x t x 2 2 t 0 with c 2 F (1.12) As a result of the softening, a discontinuity in displacement develops at x 0, with the difference in magnitude equal to 4v t L c, and strain increases infinitely with the stress dropping to zero, whilst the rest of the bar starts to unload elastically. The infinite strain in the softening domain can be expressed using the Dirac Delta function x as: x 4v t L c x (1.13) with the strain field outside the softening zone, for t L c and x 0 defined as: v x L x L x H t H t 4v t L c x c c c The solution (1.14) is symmetric for x 0. (1.14) Using the equations above, analytical solutions for displacement, strain stress and internal energy in the strain softening problem, at the response time t 3 L / 2c, are shown in Error! Reference source not found.. The key difference between the elastic solution and the strain softening (local) solution is that elastic solution provides continuous wave propagation after superposition, whereas local solution features the discontinuity in the displacements and development of the standing strain wave in the midsection of the bar. The obtained discontinuity could not propagate away from the localisation zone, due to the change of nature of the PDEs in this zone from hyperbolic to elliptic.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Consequently, material unloads outside of the localisation zone and the softening zone acts as a free boundary. Location for Figure 2 The results shown in Error! Reference source not found. will be compared to the results obtained with numerical results of the newly developed EDF model in Section 4. 3 Equivalent damage force model Our first attempt to model the strain softening problem given in Figure 1 was published in (Vignjevic, Djordjevic et al. 2014). In a series of numerical experiments, using both SPH and FEM solvers, it was shown that the size or, in the strain softening problem considered here, width of the strain softening region was controlled by the element size in classic FEM, with the strain softening localised in a single layer of elements. In the SPH, the size of the softening zone was controlled by the smoothing length, rather than the inter-particle distance, which demonstrates that the SPH method is inherently non-local and suggests that the SPH smoothing length should be linked to the material characteristic length scale in solid mechanics simulations. To address the localisation problem observed in the FEM combined with the classic CDM, an alternative approach to modelling damage localisation in FEM is proposed here. Instead of using a damage parameter, which weakens the material properties and make a negative slope of the stress strain curve, which in turn leads to imaginary wave speed in dynamic problems and changes PDEs type from hyperbolic to elliptic, damage in this model is incorporated in a form of equivalent damage force (EDF). This force is added to resultant force acting on the continuum, i.e. the right hand side of the non-homogeneous PDEs of motion so that the homogeneous part of the PDE remains unchanged relative to the elastic solution. This allows for PDEs to maintain their hyperbolic character and boundary value problem to remain well posed. The primary objective of the proposed approach is to avoid the damage induced strain softening instabilities characterised with imaginary speed of sound. For the sake of clarity, the EDF is derived below for isotropic material formulation. 3.1 Derivation of the equivalent damage force Derivation of the EDF starts from the definition of the effective stress (Kachanov 1958, Krajcinovic 1996), which is given in Equation (1.15), and calculation of the stress divergence used in the conservation of momentum (1.17) as: (1.15) 1 (1.16) Page 7 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 b a (1.17) Where: is true stress, is damage evolution variable, b is a body force vector, is material density and a is acceleration. Weak form of the conservation law (1.17) can now be derived in the Voigt notation as: d T T T w u dv w dv w dv T T T T Page 8 of 16 0 w dv w b dv w nd T T T T N N d dv B dv N dv 0 N dv N b dv N nd T T T T T T T N N d dv B dv N dv 0 T T T N dv N b dv N nd (1.18) where a standard notation for matrix of shape functions N and strain displacement matrix B was used in the expressions above, together with test function (virtual displacement vector) denoted as w. Differential equation of motion (1.20) can be rewritten as: Where the following definitions are used: T T M d Kd f f f 0 D b e (1.19) (1.20) (1.21) M N N d dv mass matrix T K B B dv stiffness matrix T T f N dv N dv D equivalent damage force T b f N b dv body force vector T e f N nd traction on a boundary

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 In this derivation, damage contributes to the conservation of momentum through the third term in Equation (1.21), f D, which requires at least one of the integrals calculated for a damaged element to be nonzero. 3.2 EDF model Implementation The model is currently developed with a scalar damage evolution function for under-integrated solid elements with a linear displacement field. This implies constant strain displacement matrix and constant strain and (effective) stress within an element. Consequently, the second term in Equation Error! Reference source not found. is equal to zero, whilst the first term in the equation is determined by damage function used. Given that the analytical nonlocal solution for strain softening problem is independent of the damage function, i.e. slope F of the stress strain curve given in Error! Reference source not found., and the fact that the wider objective of this work was to apply the model to composite materials, the EDF model was implemented together with bilinear constitutive law shown in Error! Reference source not found.. Evolution of damage is defined in terms of a single damage parameter, using a local continuum damage mechanics approach. The strain softening and the damage evolution develop when strain is in a range between i and f. Location for Figure 3 Damage and tangent stiffness, i.e. slope of the stress stain curve, for the material state determined by *, were respectively calculated as: i f 1 * E T Ei Page 9 of 16 f f i * i (1.22) (1.23) In the equations above, E is a Young modulus of undamaged material. Divergence of damage, i.e. derivative of damage parameter in the bilinear constitutive law with respect to coordinate x, can be calculated from Equation (1.22), making use of a chain rule as: x x (1.24) which is for linear elements equal to zero. This makes the first term in Equation Error! Reference source not found. and total equivalent damage force in a damaged linear solid element equal to zero. This problem can be overcome by using higher order element formulation, which provides nonzero derivatives/divergences of damage and stress tenors. Alternatively, one can calculate the divergence of damage and divergence of stress tensors numerically, which leads to the

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 nonzero EDF even for the under-integrated/linear elements. The latter approach is suitable for a nonlocal damage theory because of the advantages of constant strain elements, such as simpler numerical implementation of damage averaging compared to the higher order elements, and the fact that large higher order elements cannot represent the discontinuities due to cracking as well as the small constant strain element. Consequently, the numerically calculated EDF is adopted in this work, which is outlined below. Divergence of stress and divergence of damage in the EDF model are calculated using the following generic expression for gradient of function f x : mj I J I J, f x f x W x x l (1.25) J where indices I and J denote actual and neighbouring element, respectively, x I and corresponding coordinates, J Page 10 of 16 x J are m J and J are mass and density of the neighbouring element, W is weighting function and l is characteristic damage length, which is an input parameter for the EDF model. The EDF model is currently implemented in the LLNL Dyna3d SPH code (Liu 2004, Vignjevic, Reveles et al. 2006, Vignjevic, Campbell et al. 2009, De Vuyst, Vignjevic 2013) and the numerical results obtained with the model are given in the following section. 4 Numerical experiments The dynamic strain softening problem described in Section 2 was modelled here in a series of numerical experiments, which complement the work published in (Vignjevic, Djordjevic et al. 2014). Although the strain-softening bar problem is symmetric, the bar was discretised with odd number of elements in the loading direction, with a layer of elements/integration points in the midsection of the bar, which did not allow for application of the symmetric boundary conditions. Constitutive model defined in Error! Reference source not found. was used with three FEM solid element models shown in Error! Reference source not found., with mesh densities defined in Error! Reference source not found.. The test programme consisted of three simulation experiments: 1) experiment 1 run with mesh 1 for a constant prescribed velocity v 70 10 mm s, symmetrically applied to the free bar ends in the tensile direction, with three different damage characterisation lengths l (material input parameter); 2) experiment 2 with three mesh densities defined in Error! Reference source not found. run for applied constant prescribed velocity v 70 10 mm s, with the reference damage characteristic length l ; 3) experiment 3 run with mesh 1 and the prescribed velocity v 80 10 mm / s with the reference damage characteristic length l ; 3 3 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 The first two experiments are consistent with the numerical experiments published in (Vignjevic, Djordjevic et al. 2014) and correspond to the maximum strain developed in the midsection of the bar to be very close to the strain softening initiation (beginning of the softening behaviour), whilst the experiment 3 is chosen with the maximum strain very close to the total failure. Location for Figure 4 Location for Table 1 True stress-stain, effective stress-strain and damage-strain relationships, obtained in the impact direction with the classic CDM and the EDF approach, are given in Error! Reference source not found.. The curves were obtained with the FEM models with 101 elements along the impact direction (mesh 1) and use the same damage function defined in Equation (1.22) as illustrated in the figure. The slope of the effective stress-strain curve of the EDF model in presence of damage is equivalent to the slope of the elastic model. At the point of complete failure, which corresponds to 1, material stiffness in the EDF model drops to zero in a single step as the element is removed from the further calculation. Location for Figure 5 The simulation results of the experiment 1 for damage, displacement, strain and stress distribution, obtained at the response time t 3 L/ 2, when the stress wave propagated three quarters of the bar length, are respectively shown from Error! Reference source not found. to Error! Reference source not found.. The simulations were run with the reference mesh 1 and three values for damage characteristic lengths: l 1 3.96mm, l 2 5.94mm and l1 7.92mm. The obtained results shown a pronounced non local character, with the size of the damaged zone controlled with the damage characteristic length. Location for Figure 6 Location for Figure 7 Location for Figure 8 Page 11 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Location for Figure 9 Consequently, experiment 2 was carried out with three mesh densities and reference damage characteristic length l 3.96mm, which is an input parameter for the EDF model. The results for the damage, displacement, strain and stress distribution are shown from Error! Reference source not found. to Error! Reference source not found.. The results are stable and consistent with the nonlocal analytical solutions presented in Error! Reference source not found., with damage not localised in one layer of elements in the midsection of the bar, but distributed over a limited zone which is approximately 3l wide, as illustrated in Error! Reference source not found. and Error! Reference source not found.. Damage distribution obtained with the simulations is independent of discretisation density and is compared to the classic FEM results (Vignjevic, Djordjevic et al. 2014) from Error! Reference source not found. to Error! Reference source not found.. The maximum value of damage parameter observed in the midsection of the bar in the EDF results was equal to 0.1668, rather than 1.0, which was the local classic FEM solution (see for instance Error! Reference source not found.). Location for Figure 10 Location for Figure 11 Location for Figure 12 Location for Figure 13 Location for Figure 14 Location for Figure 15 Location for Figure 16 Location for Figure 17 Page 12 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 The third experiment was run for the impact velocity which induced almost complete failure in the midsection of the bar. The obtained results are still nonlocal and comparison of the distribution of the state variables with the results obtained in the second experiment is illustrated from Error! Reference source not found. to Error! Reference source not found.. Location for Figure 18 Location for Figure 19 Location for Figure 20 Location for Figure 21 Page 13 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 5 Conclusions It was already demonstrated in (Vignjevic, Djordjevic et al. 2014) in the series of numerical experiments that the width of the strain softening region was controlled by the element size in classic FEM and that the SPH method is inherently non-local, with the smoothing length linked to the material characteristic length scale in solid mechanics simulations. That numerical experimental programme have been complemented with the results obtained with a new EDF model proposed here, where the localisation problem of the classic FEM have been addressed by alternative definition of damage effects within the conservation of momentum equations. Performance of the EDF model was tested in a dynamic stress wave propagation problem for a range of loading cases, including the states close to the damage initiation and the states close to the complete failure. For the test cases considered, the numerical results obtained with the EDF model show stable and nonlocal character, with a reduced mesh dependency, where the size of damaged zone was controlled with damage characteristic length, which is a material input parameter. In addition, the key advantage of the EDF model is that it can be combined with any local CDM damage function. The future work on the model formulation will include the orthotropic material formulation, suitable for composite materials, which will make the model suitable for composite materials. Acknowledgement The project leading to this publication has received funding from the European Union s Horizon 2020 research and innovation programme under grant agreement No 636549. References AIFANTIS, E.C., 1992. On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science, 30(10), pp. 1279-1299. AIFANTIS, E.C., 1984. On the Microstructural Origin of Certain Inelastic Models, 1984. BAZANT, Z.P., BELYTSCHKO, T.B. and CHANG, T.-., 1984. Continuum theory for strain-softening. Journal of Engineering Mechanics, 110(12), pp. 1666-1692. BAZANT, Z.P. and BELYTSCHKO, T.B., 1985. Wave propagation in a strain-softening bar: Exact solution. Journal of Engineering Mechanics, 111(3), pp. 381-389. BAZANT, Z.P. and JIRASEK, M., 2002. Nonlocal integral formulations of plasticity and damage: Survey of progress. Journal of Engineering Mechanics, 128(11), pp. 1119-1149. BAZANT, Z.P. and ZUBELEWICZ, A., 1988. Strain-softening bar and beam: Exact non-local solution. International Journal of Solids and Structures, 24(7), pp. 659-673. DE VUYST, T. and VIGNJEVIC, R., 2013. Total Lagrangian SPH modelling of necking and fracture in electromagnetically driven rings. International Journal of Fracture, 180(1), pp. 53-70. Page 14 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 DILLON JR., O.W. and KRATOCHVIL, J., 1970. A strain gradient theory of plasticity. International Journal of Solids and Structures, 6(12), pp. 1513-1533. GRAFF, K.F., 1991. Wave motion in elastic solids. New York: Dover. HADAMARD, J.S., 1903. Cours du College de France. Lec ons sur la propagation des ondes et les e quations de l'hydrodynamique. Paris:. KACHANOV, L.M.(.M., 1958. Time of the rupture process under creep conditions. Ivz Akad Nauk SSR Otd Tech Nauk, 8, pp. 26-31. KRAJCINOVIC, D., 1996. Damage mechanics. Amsterdam ; New York: Elsevier. LEMAITRE, J., 1985. Coupled elasto-plasticity and damage constitutive equations. Computer Methods in Applied Mechanics and Engineering, 51(1-3), pp. 31-49. LIU, J., 2004. Dyna3D: A Nonlinear, Explicit, Three-Dimensional Finite Element Code for Solid and Structural Mechanics. Livermore, (CA) USA: University of California, Lawrence Livermore National Laboratory. NEEDLEMAN, A., 1988. Material rate dependence and mesh sensitivity in localization problems. Computer Methods in Applied Mechanics and Engineering, 67(1), pp. 69-85. NEILSEN, M.K. and SCHREYER, H.L., 1993. Bifurcations in elastic-plastic materials. International Journal of Solids and Structures, 30(4), pp. 521-544. PEERLINGS, R.H.J., DE BORST, R., BREKELMANS, W.A.M. and GEERS, M.G.D., 2002. Localisation issues in local and nonlocal continuum approaches to fracture. European Journal of Mechanics, A/Solids, 21(2), pp. 175-189. PEERLINGS, R.H.J., DE BORST, R., BREKELMANS, W.A.M. and GEERS, M.G.D., 1998. Wave propagation and localisation in nonlocal and gradient-enhanced damage models. Journal De Physique.IV : JP, 8(8), pp. Pr8-293-Pr8-300. PEERLINGS, R.H.J., GEERS, M.G.D., DE BORST, R. and BREKELMANS, W.A.M., 2001. A critical comparison of nonlocal and gradient-enhanced softening continua. International Journal of Solids and Structures, 38(44-45), pp. 7723-7746. PIJAUDIER-CABOT, G. and BAZANT, Z.P., 1987. Nonlocal damage theory. Journal of Engineering Mechanics, 113(10), pp. 1512-1533. PIJAUDIER-CABOT, G., BAZANT, Z.P. and TABBARA, M.R., 1988. Comparison of various models for strain softening. Engineering Computations, 5(2), pp. 141-150. RUDNICKI, J.W. and RICE, J.R., 1975. Conditions for the localization of deformation in pressuresensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23(6), pp. 371-394. SLUYS, L.J. and DE BORST, R., 1994a. Dispersive properties of gradient-dependent and ratedependent media. Mechanics of Materials, 18(2), pp. 131-149. Page 15 of 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 SLUYS, L.J. and DE BORST, R., 1994b. Dispersive properties of gradient-dependent and ratedependent media. Mechanics of Materials, 18(2), pp. 131-149. SLUYS, L.J. and DE BORST, R., 1992. Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples. International Journal of Solids and Structures, 29(23), pp. 2945-2958. TVERGAARD, V. and NEEDLEMAN, A., 1997. Nonlocal effects on localization in a void-sheet. International Journal of Solids and Structures, 34(18), pp. 2221-2238. TVERGAARD, V. and NEEDLEMAN, A., 1995. Effects of nonlocal damage in porous plastic solids. International Journal of Solids and Structures, 32(8-9), pp. 1063-1077. VIGNJEVIC, R., CAMPBELL, J., JARIC, J. and POWELL, S., 2009. Derivation of SPH equations in a moving referential coordinate system. Computer Methods in Applied Mechanics and Engineering, 198(30 32), pp. 2403-2411. VIGNJEVIC, R., DJORDJEVIC, N., GEMKOW, S., DE VUYST, T. and CAMPBELL, J., 2014. SPH as a nonlocal regularisation method: Solution for instabilities due to strain-softening. Computer Methods in Applied Mechanics and Engineering, 277, pp. 281-304. VIGNJEVIC, R., REVELES, J.R. and CAMPBELL, J., 2006. SPH in a total lagrangian formalism. CMES - Computer Modeling in Engineering and Sciences, 14(3), pp. 181-198. VON KARMAN, T. and DUWEZ, P., 1950. The propagation of plastic deformation in solids. Journal of Applied Physics, 21(10), pp. 987-994. Page 16 of 16

Figure Figure 1 a) Geometry and loading of softening bar; b) stress strain behaviour (Bazant, Belytschko 1985)

Figure a) b) c) d) Figure 2 Elastic local and nonlocal solutions at response time t=3l/2c for: a) normalised displacement; b) normalised strain; c) normalised stress; d) normalised internal energy;

Figure Figure 3 Bilinear law implemented in the FEM and SPH codes using a damage parameter and classic CDM approach

Figure Figure 4 Spatial discretisation of the dynamic strain-softening bar

Figure Figure 5 Stress strain and damage strain relationships obtained using CDM and EDF models; mesh 1 model with 101 elements along the impact direction

Figure Figure 6 Damage calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three simulations run with three damage characteristic lengths

Figure Figure 7 Longitudinal displacement calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three simulations run with three damage characteristic lengths

Figure Figure 8 Longitudinal strain component calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three simulations run with three damage characteristic lengths

Figure Figure 9 Longitudinal stress component calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three simulations run with three damage characteristic lengths

Figure Figure 10 Damage calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three mesh densities run with the same damage characteristic lengths

Figure Figure 11 Longitudinal displacement calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three mesh densities run with the same damage characteristic lengths

Figure Figure 12 Longitudinal strain component calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three mesh densities run with the same damage characteristic lengths

Figure Figure 13 Longitudinal stress component calculated along the bar model in experiment 1 at response time t 3 L/ 2 ; three mesh densities run with the same damage characteristic lengths

Figure a) b)

c) Figure 14 Nonlocal damage distribution obtained with three FEM models with constant characteristic damage length: a) mesh 1; b) mesh 2; c) mesh 3;

Figure a) b) Figure 15 Damage distribution obtained with a bar model with 101 linear elements along impact direction: (a) classic FEM; (b) EDF model;

Figure a) b) Figure 16 Damage distribution obtained with a bar model with 151 linear elements along impact direction: a) classic FEM; b) EDF model;

Figure a) b) Figure 17 Damage distribution obtained with a bar model with 201 linear elements along impact direction: a) classic FEM; b) EDF model;

Figure Figure 18 Damage parameter calculated along 101 element bar model in experiment 2 and experiment 3 at the time instant t 3 L/ 2

Figure Figure 19 Longitudinal displacement calculated along 101 element bar model in experiment 2 and experiment 3 at the time instant t 3 L/ 2

Figure Figure 20 Longitudinal strain component calculated along 101 element bar model in experiment 2 and experiment 3 at the time instant t 3 L/ 2

Figure Figure 21 Longitudinal stress component calculated along 101 element bar model in experiment 2 and experiment 3 at the time instant t 3 L/ 2

Table Table 1 Mesh density used in the FEM models of strain softening problem Number of elements Impact direction x In-plane y In-plane z Mesh 1 101 5 5 Mesh 2 151 7 7 Mesh 3 201 10 10