Numerical Erosion in Continuum Damage Mechanics. Shashank N Babu. Master of Science Thesis

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1 Numerical Erosion in Continuum Damage Mechanics

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3 Numerical Erosion in Continuum Damage Mechanics in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering at the Delft University of Technology, January 2, 216 Faculty of Civil Engineering and Geosciences Delft University of Technology

4 Copyright c Faculty of Civil Engineering and Geosciences (CitG) All rights reserved.

5 Delft University of Technology Department of Faculty of Civil Engineering and Geosciences (CitG) The undersigned hereby certify that they have read and recommend to the Faculty of Civil Engineering and Geosciences for acceptance of a thesis entitled Numerical Erosion in Continuum Damage Mechanics by in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering Dated: January 2, 216 Supervisor(s): Prof. Dr. Ir. L.J. Sluys Dr. Ir. J. Weerheijm L.F. Magalhaes Pereira

6 VI Ir. L.J.M. Houben

7 Table of Contents Table of Contents List of Figures List of Tables Acknowledgements Glossary List of Acronyms List of Symbols i iii v vii ix ix ix 1 INTRODUCTION 1 2 LITERATURE REVIEW INTRODUCTION TO WAVE PROPAGATION DAMAGE MODELS Isotropic damage models Standard nonlocal damage models Stress based nonlocal damage model NUMERICAL EROSION DAMAGE MODELS PROBLEM CASES LOADING PARAMETERS LOCAL DAMAGE MODEL NONLOCAL (NL) DAMAGE MODEL STRESS BASED NON LOCAL (SBNL) DAMAGE MODEL Numerical stability NL VS SBNL DAMAGE MODEL SUMMARY

8 ii TABLE OF CONTENTS 4 EROSION IN LOCAL DAMAGE MODEL STRESS BASED EROSION LIMIT STRAIN BASED EROSION LIMIT DAMAGE BASED EROSION LIMIT SUMMARY EROSION IN NONLOCAL MODELS NONLOCAL COMPUTATION SCHEME EROSION IN STANDARD NONLOCAL MODEL Strain based erosion limit Nonlocal variable based erosion limits Damage based erosion limit EROSION IN STRESS BASED NONLOCAL MODEL Strain based erosion limit Nonlocal variable based erosion limit Damage based erosion limit SUMMARY ERRONEOUS INFLUENCE DOMAIN & CORRECTION CONCEPTUALISATION OF THE PROBLEM DAMAGE BASED EROSION LIMIT (ω =.8) INFLUENCE DOMAIN CORRECTION INFLUENCE OF MESH REFINEMENT SUMMARY CONCLUSION AND RECOMMENDATIONS CONCLUSION PHYSICAL SIGNIFICANCE AND RECOMMENDATIONS References 57

9 List of Figures 1-1 Main failure mechanism in concrete subject to dynamic load Wave reflection in elastic medium from boundaries [2] Degradation of the material stiffness Interaction domain in standard nonlocal models Influence of a distributing point (Gaussian function) [1] Schematic of cantilever beam under study Schematic of spalling failure under study Stress-strain curve for the material used Loading profile for the cases under study Results using the local model Damage profile with l int = 1 mm Damage profile for case Damage profile for case Nonlocal strains for case Numerical instabilities in case 2 with ρ min = Damage profile for case 1 with ρ min = Damage profile for case 2 with ρ min = Nonlocal strains for case 2 with ρ min = SBNL vs NL damage model SBNL vs NL damage model Post-peak behaviours observed in material Erosion with limiting σ max = 3. MPa Strain values used for erosion limit

10 iv LIST OF FIGURES 4-4 Erosion in case 1 with limiting ε max = Erosion in case 2 with limiting ε max = Erosion with limiting ω = Computation scheme used in the code Results with limiting ε max =3 1 4 (κ NLV) Erosion in case 1 (κ NLV) with limiting ε max = Erosion in case 2 (κ NLV) with limiting ε max = Evolution of principal strain (type A) post-erosion in case Erosion in case 1 (κ NLV) with limiting κ= Erosion in case 2 (κ NLV) with limiting κ= Erosion in case 2 (ε eq NLV) with limiting ε eq = Erosion in case 1 with limiting ω= Erosion in case 2 with limiting ω= Erosion in case 1 (κ NLV) with limiting ε max = Erosion in case 2 (κ NLV) with limiting ε max = Damae profile with limiting κ= Damage profiles for case 2 with limiting NLV = Erosion in case 1 with limiting ω= Erosion in case 2 with limiting ω= Influence domain in the nonlocal models Damage profile in case 1 with limiting ω= Damage profile in case 2 with limiting ω= Representation of the deleted zones (κ NLV) Evolution of nonlocal variable in case 1 with limiting ω= Damage profile with one element deletion with ω= Damage profile in case Damage profile in case 2 (κ NLV) post-correction with deletion at ω=.8. 51

11 List of Tables 3-1 Material Properties Nonlocal model properties Nonlocal model properties Element deletion in case 1 and Element deletion in case 2 (κ NLV) Element deletion with κ as the NLV Element deletion with κ as the NLV Element deletion with ε eq as the NLV Element deletion in case Element deletion in case Element deletion in case 1 for both nonlocal variables Element deletion in case Element deletion in case Element deletion in case Element deletion in case Erosion models summary Element deletion in case Element deletion in case Element deletion in case

12 vi LIST OF TABLES

13 Acknowledgements I would like to express my sincere gratitude to Technische Universiteit Delft for letting me fulfil my dream of being a student here. I would also like to thank my parents for supporting and helping me in every phase of my life. I would like to thank my supervisor Prof. Dr. Ir. L.J. Sluys, Dr.ir. J. Weerheijm, Ir. L.J.M. Houben and L.F. Magalhaes Pereira for their assistance and patience during the course of this study. Their support is special as they were extremely understanding when I met with several drawbacks during the course of this thesis. Lastly, a big thank you to all the people in Delft who made my stay away from home memorable and unforgettable. Special mention to Adarsha, Ajay Prasad, Avinash Suresh, Cindhuja, Dharani, Foram, Gowrishankar, Harshal, João, Karthik, Manimaran, Sajjad, Srinidhi, Srivatsan and Venky for all the fun times, good company and their unconditional friendship especially during the bad times. Delft, University of Technology January 2, 216

14 viii Acknowledgements

15 Glossary List of Acronyms FPZ el NL SBNL NLV Fracture Process Zone Element Standard Nonlocal Model Stress Based Nonlocal Model Nonlocal Variable List of Symbols α σ β κ κ κ f ν ω φ ρ ρ min σ σ max κ ε eq Exponential Softening Control Parameter Effective Stress Exponential Softening Control Parameter Strain History Variable Strain at Damage Initiation Ultimate Failure Strain Poisson s Ratio Damage Variable Weight Function Internal Length Control in Stress based Nonlocal Model Ratio between Internal Length and Minimum Internal Length Stress Maximum Principal Stress Nonlocal Strain History Variable Nonlocal Equivalent Strain

16 x Glossary ε ε eq ε max D e D s f t l int l min t load t run A E exp f G I J K k L t Strain Equivalent Strain Maximum Principal Strain Elastic Stiffness Matrix Secant Stiffness Matrix Tensile Strength Internal Length Minimum Internal Length Load Duration Time Simulation Run Time Cross-sectional Area Elastic Modulus Exponential Function Loading Function Shear Modulus First Invariant Second Invariant Bulk Modulus Ratio between uniaxial compressive and uniaxial tensile strength of concrete Length Time

17 Chapter 1 INTRODUCTION Concrete is a quasi-brittle material used in a multitude of construction practices. Common examples to these practices include high rise sky scrappers, bridges, residential buildings and military infrastructure. Some of these structures, especially the military ones, are required to withstand large stresses and strains which are a consequence of blast loading (dynamic). In order to facilitate a safe design of such structures, an accurate description of the response of the material to dynamic loads is required. Concrete structures under blast load can exhibit different failure mechanisms as shown in fig Cratering and spalling are the major failure mechanisms observed and are caused by compressive and tensile failure respectively. Other failure types include scabbing and perforation. Cratering Spalling Figure 1-1: Main failure mechanism in concrete subject to dynamic load. Upon detonation, a high hydrostatic compressive state is developed causing irreversible compaction and crushing which leads to cratering. The compressive wave propagates through the structure and reaches the inside. Reflection from the free end causes propagation of tensile wave which leads to spalling failure and is analysed in this study. Commonly used practices to assess the performance of concrete structures are either experimental methods, where the actual situation is mimicked, or, numerical simulations which can be related to the response observed in existing cases. Experimental methods are possible for limited scenarios, are expensive and require elaborate preparations while numerical simulations are a feasible option for most cases and are cost effective.

18 2 INTRODUCTION Numerical simulation using continuum damage mechanics is utilised to predict the fracture process zone (FPZ) formed. This is done using a constitutive relation describing the evolution of damage which represents the degradation of stiffness matrix. Modelling spalling failure is accomplished by utilizing three models which already exist in literature, namely the isotropic damage model (local damage model) and two nonlocal models. Local models are documented to suffer from spurious mesh dependence and localisation of strains. To rectify this, the two integral nonlocal models are used. The study is restricted to standard and stress based nonlocal damage model [1]. Both these models introduce the concept of internal length, over which the local strain is averaged by means of a weight function. These models are implemented as a user defined material model in the framework of a commercially available finite element package named LS-DYNA, which is well suited for numerical simulation of wave propagation in a continuum. It uses an explicit computation scheme where the critical time step is decided using Courant condition. Generally, Lagrangian codes are used for solids while Eulerian codes are used to study fluids. Lagrangian meshes used to model cratering or spalling show excessive deformation or reach a completely damaged state (zero stiffness) respectively. This leads to destabilisation of the numerical simulation. In the case of excessive deformation, critical time step calculated based on Courant condition becomes low, thus burning up excessive computational resources with little progress towards the final result. In tensile failure, a large number of damaged elements (zero stiffness), when considered in the computational sequence, can lead to instabilities. Numerical erosion can be of vital importance in both cases. This tool is available in most commercial packages and selectively deletes elements that exceed a particular erosion limit defined by the user. This study aims at understanding the use of this tool while predicting spalling failure in concrete. The primary objective of this study is to: Assess different erosion limits for their robustness, mesh sensitivity and physical significance. Couple the relevant erosion limits with the models to further improve them. While numerical erosion is a sensible approach to stabilize the simulations, it is an irreversible process. It is also associated with a new set of anomalies which require attention. Deleting elements causes loss of energy and mass. Analysing the influence of these on the results of the simulation are out of the scope of this study.

19 Chapter 2 LITERATURE REVIEW This chapter provides a theoretical overview of the pre-requisites for the numerical simulations used to predict failure. The first section presents the behaviour of waves in elastic medium, the second section introduces the damage models used in this study and the final section presents a detailed overview on numerical erosion. 2.1 INTRODUCTION TO WAVE PROPAGATION Understanding the basic principles of wave propagation in an elastic continuous solid media is important as they form an integral part in understanding the response of concrete to dynamic loads. Spalling failure (fig. 1-1) occurring during an explosion is characterised as tension failure and is caused by reflection of the compressive wave encountering a free boundary. Stress Displacement/Velocity Stress Displacement/Velocity (a) Reflection off Fixed Boundary (b) Reflection off Free Boundary Figure 2-1: Wave reflection in elastic medium from boundaries [2] The word reflection in this study, refers to the behaviour of a certain wave (stress, velocity or displacement) when it encounters a boundary. Reflection of a wave from a fixed boundary

20 4 LITERATURE REVIEW causes a stress wave to double in amplitude and remain in the same state as the incident wave while a velocity or displacement wave changes direction (positive to negative, and vice-versa) thereby resulting in a net zero condition at the fixed end (fig. 2-1a). Reflection of a wave from a free boundary causes a change in state (compressive to tensile, and viceversa) for a stress wave resulting in a net zero condition at the free end while velocity or displacement wave doubles in amplitude and remains in the same state as the incident wave (fig. 2-1b). 2.2 DAMAGE MODELS Damage mechanics is suitable for making engineering predictions about the initiation and propagation of fracture in materials without resorting to a microscopic description that would be too complex for practical engineering analysis [3]. Failure in concrete can be numerically modelled by damage. One concept to model damage is the continuum damage mechanics, a constitutive theory describing the progressive loss of material stiffness or strength due to deterioration in the form of propagation and coalescence of micro-cracks, micro-voids, and similar defects [4]. Continuum damage mechanics has been successfully used to simulate failure in concrete, especially tensile failure. It is also less expensive and easier to implement when compared to other approaches (example: plasticity and discrete crack approach). σ D e D s Figure 2-2: Degradation of the material stiffness. ε Damage is represented by a scalar parameter ω and it directly captures the loss of stiffness as shown in fig Evolution of damage in the zone termed Fracture Process Zone (FPZ) determines the direction of crack propagation [5]. Evolution of damage in the material depends on the post-peak behaviour of the material. Post-peak strain softening is the deciding factor in representation of a quasi-brittle material such as concrete. Different types of softening laws are available and common examples are linear and exponential.

21 2.2 DAMAGE MODELS Isotropic damage models Damage, controlled by a scalar parameter (ω), has a value between (virgin material) to 1 (completely damaged material). Loss of stiffness as damage evolves is incorporated into the stiffness matrix as shown in equation 2-1. It is also a direct representation of the stress state of the material (equation 2-2). D s = (1 ω)d e (2-1) σ = D s : ε = (1 ω)d e : ε (2-2) Where, D s = Damaged stiffness matrix, D e = Undamaged stiffness matrix, σ = Stress tensor, ε = Strain tensor The evolution of ω depends on equivalent strain which is characterised by the strain state of the material. Equation 2-3 gives the modified von Mises strain used in this study. Where, ε eq = (k 1)I 1ε 2k(1 2ν) + 1 (k 1)2 2k (1 2ν) 2 I2 1ε + 12kJ 2ε (2-3) (1 + ν) 2 k = Ratio between uniaxial compressive and uniaxial tensile strength of concrete, J 2ε = Second deviatoric strain invariant, I 1ε = First strain invariant, ν = Poisson s ratio. A loading function (equation 2-4) is introduced to specify the elastic domain of the material. It depends on the equivalent strain (ε eq ), scalar parameter κ, and is used to control the evolution of the elastic domain. The history variable κ stores the maximum equivalent strain reached in the simulation. Damage evolution is governed by Kuhn-Tucker loadingunloading conditions given by equation 2-5. f(ε, κ) = ε eq (ε) κ (2-4) f, κ, κf = (2-5)

22 6 LITERATURE REVIEW This study uses exponential softening law as it is physically closer to the behaviour observed in concrete [6]. Damage variable using this law is given by equation 2-6. κ is based on the elastic domain and κ f is based on the material under study. Where, ω = 1 κ κ [ exp ( κ κ κ f )] κ = History parameter recording the peak equivalent strain, κ = Strain at which the damage initiates, κ f = Ultimate strain at failure (2-6) Standard nonlocal damage models. It is vastly documented that the isotropic (local) damage models have several limitations such as spurious mesh dependence, localisation of strain and unreliable damage profiles [4]. Many modifications to this approach have been proposed over the years to minimize these limitations, with nonlocal regularisation techniques becoming commonly used methods. Standard nonlocal models are among the most successful regularisation techniques [8]. A nonlocal continuum is a continuum in which some state variable depends not only on the stresses or strains at the same point but also on the stress or strain field in the neighbourhood of the point. In quasi-brittle materials, nonlocality originates in the interactions between micro-cracks and leads to stress magnification in the areas surrounding a micro crack, thus allowing for the creation or growth of micro cracks [7]. This nonlocal behaviour is introduced by means of an internal length (l int ), over which a particular variable is averaged using a weight function, thus offering physically reliable results by explicitly introducing the nonlocal nature of micro-cracking. l int x Weight function s Figure 2-3: Interaction domain in standard nonlocal models.

23 2.2 DAMAGE MODELS 7 A schematic representation of the interaction domain can be seen in fig The target element (x) is influenced by the elements which surround it (at s) and is computed by using a weight function with the internal length specifying the maximum interaction domain. Several nonlocal versions have been proposed over the years to simulate failure in concrete. Pijaudier and Bažant [8] suggested that one could average the damage parameter ω computed and obtain a nonlocal parameter, which resulted in unsatisfactory results. Pijaudier and Bažant [9] also suggested averaging the damage energy release rate and evaluate the damage parameter corresponding to the maximum previously reached value of the nonlocal damage energy. While in the elastic state, it remains zero and the response is linear elastic. Numerical implementation of the nonlocal damage model based on averaging of equivalent strain is relatively straightforward. The evaluation of stresses from given strains remains fully explicit, and no internal iteration loop is needed [4]. In this study, equivalent strain (or another parameter, κ) given by equation 2-3 is replaced with an nonlocal equivalent strain ε eq or κ over a volume Ω by means of an internal length l int. The response of an element is now dependent on its neighbouring elements. The averaging of a local variable involves a weight function φ(x s) defining the interaction between x (target point) and its neighbouring point located at s inside the volume Ω (equation 2-7). Ω ε eq (x) or κ(x) = φ(x s) (ε eq or κ) ds Ω φ(x s) ds (2-7) Gauss distribution is the choice of the weight function used in this study and is given by equation 2-8. ( ) 2 2 x s φ(x s) = exp (2-8) l int Stress based nonlocal damage model Although the standard nonlocal model successfully solves the mesh dependence of the local damage model, it has several drawbacks. At total failure, the interaction is still active, thus leading to diffusion of damage. This constant description of the nonlocal domain also leads to inadequate treatment of interaction near the vicinities of the free boundary and newly formed discontinuities post-damage [1]. The stress based nonlocal model is an improved regularisation technique proposed by Giry et al. [1] to rectify the shortcomings of the standard nonlocal model. It no longer considers what a point located at x can receive, but instead what a point located at s can distribute [1]. The nonlocality is defined as a quantity given by each point located at s along its principal stress direction with an intensity depending on the level of principal stress. This facilitates a dynamic influence domain as shown in fig The stress field allows for a direct description of the presence of a free boundary and the development of FPZ, thus treating the newly formed boundaries more realistically than the standard nonlocal model [1].

24 8 LITERATURE REVIEW (a) Standard non-local model (b) Stress based non-local model Figure 2-4: Influence of a distributing point (Gaussian function) [1]. Evolution of the nonlocal domain is controlled by a scalar parameter (ρ) that is multiplied with the internal length (l int ). This parameter has a value for unloaded specimen and 1 when approaching the tensile strength (f t ) of the material. The ellipse or the influence domain is centred at s and has its major axis aligned with the principal stress direction as shown in fig The length of the axis is determined by the scalar parameter ρ l int. The resultant domain calculation has two parts: firstly, the calculation of the interaction domain of each integration point, and secondly, the definition of the influence domain from these interaction domains. The new influence domain calculation is done using the stress state at the points from the previous step. This is done for every time step unlike the standard nonlocal approach and is computationally expensive. For the standard nonlocal approach, the interaction domain is chopped off or truncated at the boundaries leading to loss of symmetry of the weight function in the vicinities of a boundary [1]. In the stress based non-local approach, the weight function is modified as shown in equation 2-9. The resulting domain may or may not be symmetric, (i.e. φ(x s) φ(s x)). ( ) 2 2 x s φ(x s) = exp (2-9) ρ l int The influence domain can now shrink or expand based on the stress state of each element. Numerical instabilities occur when it shrinks to zero. To prevent this, minimum internal length (l min ) is set by the user which is used to obtain the value of ρ min. ρ min = l int l min (2-1) The value of ρ min is utilised in this study to assess the numerical stability of stress based nonlocal model.

25 2.3 NUMERICAL EROSION NUMERICAL EROSION Erosion is a term that is normally heard and understood in its physical form which means gradual destruction or diminution. Owing to this unfortunate choice of terminology, differentiating the form of erosion used in this study from its physical form is needed. This nomenclature may also lead to a case, where one may assume numerical erosion mimics the actual physical erosion. This part may or may not be true based on how the tool is implemented and also the material under study. Erosion in this study refers to a numerical tool that deletes elements (along with the attached nodes) which have reached a particular limit. The general consensus on erosion is to achieve element deletion without compromising the accuracy of the results. The primary objective of this study is to assess the different limits available for their robustness, mesh dependence and physical significance using the models implemented to predict tensile failure in concrete. To understand its importance, an introduction to hydrocode and its types is necessary. Hydrocode is a computational tool for modelling the behaviour of continuous media. In its purest sense, it is a computer code for modelling fluid flow at all speeds. It can, however, be adapted to treat material strength and a range of rheological models for material behaviour [11]. Two main types are distinguished, Lagrangian codes and Euler codes. Other types are a hybrid of both codes like the arbitrary Euler Lagrange codes (ALE) or a meshless (free-lagrange) method [12]. Lagrangian codes have dominated the world of explicit codes. In this study, LS-Dyna is used with Lagrangian meshes and an explicit computation scheme. Stability of an explicit numerical scheme is driven by the time step that is chosen. The largest appropriate time step size is limited by the Courant condition on sound signal propagation [11]. This time step is the minimum time taken for sound to cross an element. This time step is then usually scaled down further to improve numerical stability. These codes have Lagrangian system/mesh that define the geometry of the model. A grid is embedded with the material. Lagrange codes track the flow of individual masses and this grid deforms together with the material [12]. One main problem with Lagrange meshes is large distortions. As the grid distorts with the material, element size tends to reduce. When an explicit numerical scheme is used, the calculation of the next time step is based on the size of the smallest element in the grid. When the element size tends to zero, the time step tends to zero, due to which, little progress is made with each time step leading to excessive use of computational resources [12]. In tensile failure, a rapid degradation of the material stiffness is observed. A large number of damaged elements in the computational sequence can lead to numerical instabilities. Numerical erosion is a possible solution for both compressive and tensile failure. Erosion algorithms are used to selectively remove elements. However, they do not conserve energy or mass [12]. The mass and energy of the deleted element is either removed or distributed to the corner nodes of the adjacent element. In LS-DYNA, the default option is deletion of mass upon erosion and is used in this study.

26 1 LITERATURE REVIEW Formulation of the best erosion limit for a particular simulation is extremely difficult. These limits are usually adjusted based on a particular simulation in consideration with the constitutive law, loading and material under study. An excerpt from Introduction to Hydrocodes [12] is quoted to understand the importance of the limit: Though element removal (erosion) associated with total element failure has the appearance of physical material erosion, it is, in fact, a numerical technique used to permit extension of the computation. Without numerical erosion, severely crushed elements in Lagrangian calculations would drive the simulation time step to a very small value, resulting in the expenditure of very many computational cycles with negligible advance in the simulation time. Also, Lagrangian elements which have become very distorted have a tendency to lock up, thereby inducing unrealistic distortions in the computational mesh. If the erosion strain is set too low, there is unrealistic excessive removal of material. If it is set too high, grid lockup and minimum time step violations can occur.. This indicates that formulation of a limiting value for an erosion limit should be done with extreme caution as the process is irreversible. Commercial packages generally recommend that this limit is set after calibration with experimental results. Different types of limits available in most commercial packages are summarised below: Stress Based: There are different stress based erosion limits such as pressure, principal stress, effective stress etc. based on either the commercial software or the code. Strain Based: There are different strain based erosion limits such as equivalent strain, principal strain, plastic strain etc. based on either the commercial software or the code. Damage Based: Damage is a monotonically increasing variable which represents the physical state of the material by a scalar measure and can be also be used as an erosion limit. It is not commonly provided with many commercial softwares. The strain based limits are most commonly used [13]. In this study, robustness, mesh dependence and physical significance of the different erosion limits are validated with the models used. This is done by using the erosion algorithm provided in LS-DYNA as well as a custom erosion algorithm implemented by the authors.

27 Chapter 3 DAMAGE MODELS Two one dimensional cases are analysed to facilitate a backbone for implementation of numerical erosion. The results of the numerical simulations using the damage models that are discussed in section 2.2 are presented in this chapter. 3.1 PROBLEM CASES As numerical erosion is not widely used and since there is limited literature, it is decided to simplify the failure observed in concrete under impact loads into a one dimensional premise. In this study, elements of the mesh are numbered from left to right as indicated in fig. 3-1 and 3-2. Any reference to a particular element is done using el(x) where x is the element number and el represents element. The two cases studied are: Case 1:- Tensile failure in a cantilever beam. A beam fixed at the left end with a tensile stress pulse incident at the right end. Tensile failure is observed at the fixed end after reflection. x P(t) P L Figure 3-1: Schematic of cantilever beam under study. t 1 t Case 2:- Spalling failure in a beam. A beam free on both ends is loaded with a compressive stress pulse. Free end reflection causes a change in stress state leading to tensile failure.

28 12 DAMAGE MODELS P(t) x t 1 t 2 t load t L P Figure 3-2: Schematic of spalling failure under study. In both cases, the beam has a length (L) of 1 mm and cross sectional area (A) of 1 mm 2. Table 3-1 shows the material properties used for the simulation. Table 3-1: Material Properties Material Properties Value Units Density 24 kg m 3 Elastic Modulus (E) N mm 2 Shear Modulus (G) N mm 2 Bulk Modulus (K) N mm 2 Tensile Strength (f t ) N mm 2 Poisson s Ratio (ν) - In both cases, the beam is modelled using 8-node solid element with one integration point and the number of elements vary from 5 to 5 along the length. Stress strain relationship (using exponential softening law) of the material described in table 3-1 is shown in fig This is obtained by simulating a cube subject to a tensile stress pulse. The values of κ and κ f are and respectively (fig. 3-3). 3 σ x (MPa) Strain (ε x ) Figure 3-3: Stress-strain curve for the material used.

29 3.2 LOADING PARAMETERS LOADING PARAMETERS Loading and other simulation related parameters are shown in fig. 3-4a and 3-4b for case 1 (cantilever beam) and case 2 (spalling failure) respectively. For case 1, the linear domain of the stress pulse extends till s (t 1 ) and continues to stay constant till the simulation time (t run = s) after reaching 2 MPa. σ (MPa) t Time (1-6 ) (s) (a) Tensile pulse used in case 1. σ (MPa) t 1 t 2 t load Time (1-6 ) (s) (b) Compressive pulse used in case 2. 5 Figure 3-4: Loading profile for the cases under study. For case 2, a trapezoidal stress pulse is used with load time (t load ) of s and linear domain at ± s from start (t 1 ) and end (t 2 ) respectively with a simulation time (t run ) of s.

30 14 DAMAGE MODELS 3.3 LOCAL DAMAGE MODEL The local damage model is used to predict failure in both cases under study. Fig. 3-5 shows highly mesh sensitive and unreliable results. The width of the FPZ predicted can be related to the failure zone developed in the material. However, it offers limited physical significance in both cases because the damaged zone does not represent a continuum i.e., multiple localised damage peaks are predicted in both cases. Damage profile for case 2 from a 5 elements mesh (fig. 3-5c) is shown to illustrate the extent of localisation with mesh refinement in the local damage model. The heavy extent of localisation observed in the results obtained is due to dynamic nature of the problem and the explicit computation scheme used by LS-DYNA. The classical solution to these problems gives a damage zone localised to a single element [14] El 75 El 1 El El 75 El 1 El (a) Damage profile for case 1. (b) Damage profile for case El (c) Damage profile in case 2 with fine mesh. Figure 3-5: Results using the local model.

31 3.4 NONLOCAL (NL) DAMAGE MODEL NONLOCAL (NL) DAMAGE MODEL The standard nonlocal damage model is used to predict failure in both cases under study. Fig. 3-6 shows the results for 1 mm internal length for both cases and it is clearly seen that there is excessive damage diffusion with κ as the nonlocal variable. This indicates that the internal length is not entirely material dependent. This study categorises strain into two types. Type A strain variable is defined such that they can increase or decrease during the simulation. All traditional strain quantities such as equivalent strain, principal strain etc. come under this category. Type B strain variables are those that grow monotonically. Strain history variable κ comes under this category. History variable κ (type B) continues to contribute to its surroundings due to the constant description of the interaction domain in the nonlocal model, thus predicting the excess damage. 1.5 NLV κ NLV εeq (a) Case NLV κ NLV εeq (b) Case 2. Figure 3-6: Damage profile with l int = 1 mm. After a sensitivity study, it is decided to use different values for internal length based on the nonlocal variable used (table 3-2). Internal length of 1 mm for κ and 15 mm for ε eq are chosen as they resulted in similar spread of damage at the peak levels. This clearly establishes that the internal length depends on the material and also on the choice of the nonlocal variable for the standard nonlocal model. Table 3-2: Nonlocal model properties Nonlocal Variable Model parameter ε eq κ l int 15 mm 1 mm Fig. 3-7 and 3-8 indicate that the nonlocal technique solves the spurious mesh dependence observed in the local damage model used in this study. Mesh independent results were obtained irrespective of the choice of the nonlocal variable.

32 16 DAMAGE MODELS El 25 El 5 El Damage ω El 25 El 5 El (a) κ as the NLV (b) ε eq as the NLV. Figure 3-7: Damage profile for case El 25 El 5 El El 25 El 5 El (a) κ as the NLV (b) ε eq as the NLV. Figure 3-8: Damage profile for case 2. An interesting observation in case 2 is that there is slight localisation (fig. 3-8a) in the damage profile with κ as the nonlocal variable. This is not seen with ε eq as the nonlocal variable (fig. 3-8b). There is a slight increase in the width of the FPZ obtained due to the increased internal length and this is deemed acceptable as the increase is less than 5% for both cases. It also solves the slight localisation seen with κ (nonlocal variable). These variables produce minor variations in the damage profiles predicted and the maximum nonlocal strains they compute also vary significantly. There is a decrease in the value of nonlocal strain when ε eq is used as the nonlocal variable (fig. 3-9). This is due to the increase in the internal length resulting in a larger zone to average the local strains thereby reducing the amplitude of the nonlocal strains.

33 3.5 STRESS BASED NON LOCAL (SBNL) DAMAGE MODEL κ El 25 El 5 El (a) κ profile over the length. ε El 25 El 5 El (b) ε eq profile over the length. Figure 3-9: Nonlocal strains for case STRESS BASED NON LOCAL (SBNL) DAMAGE MODEL The stress based nonlocal damage model is used to predict failure in both cases under study. In this model, there is an additional model parameter called minimum internal length (l min ) which is controlled by the scalar variable ρ min. The variation of l min influences the results in the stress based nonlocal model and needs to be checked for numerical stability. Therefore, ρ min is varied from. to.4 (with increments of.1) in search of numerically stable results Numerical stability A sensitivity study is conducted using the varying values of ρ min in search for mesh independent and numerically stable results. It is observed that the variation in the minimum internal length did not influence the results of case 1 as much as it did in case 2. Low values of ρ min produced numerical instabilities only visible in case 2. Results with ρ min =.2 is shown in fig It is observed that there is some damage propagation towards the left (fig. 3-1) while using both κ and ε eq as the nonlocal variable and some early localisations at the peak damage levels in case of ε eq. It is important to note that the width of the FPZ predicted in each case in the stress based nonlocal model was identical with 1 mm internal length for both nonlocal variables. However, the values used in the standard nonlocal model are set even in the stress based nonlocal model. This is because ε eq (nonlocal variable) with 1 mm internal length produced numerical instabilities (localisations) as seen in fig. 3-1b.

34 18 DAMAGE MODELS El 25 El 5 El El 25 El 5 El (a) Damage profile with κ as the NLV (b) Damage profile with ε eq as the NLV. Figure 3-1: Numerical instabilities in case 2 with ρ min =.2. As the value of ρ min equals.3 (table 3-3), numerically stable and mesh independent results are obtained as shown in fig and Both κ and ε eq as the nonlocal variable resulted in numerically stable results for both cases. Table 3-3: Nonlocal model properties Model parameter Nonlocal Variable ε eq κ l int 15 mm 1 mm l min 4.5 mm 3 mm El 25 El 5 El El 25 El 5 El (a) κ as the NLV. (b) ε eq as the NLV. Figure 3-11: Damage profile for case 1 with ρ min =.3.

35 3.5 STRESS BASED NON LOCAL (SBNL) DAMAGE MODEL 19 Results obtained using this model also produce a similar increase in the width of the FPZ predicted with ε eq as the nonlocal variable. It also overcame the slight localisations (fig. 3-12b) that were seen when using κ as the nonlocal variable (fig. 3-12a). This is in concurrence with the observation made in section 3.4 and the increased FPZ is deemed acceptable because both nonlocal variables produce identical damage plateau at peak level El 25 El 5 El El 25 El 5 El (a) κ as the NLV. (b) ε eq as the NLV. Figure 3-12: Damage profile for case 2 with ρ min = El 25 El 5 El.5 1 El 25 El 5 El κ -.35 ε (a) κ profile over the length (b) ε eq profile over the length. Figure 3-13: Nonlocal strains for case 2 with ρ min =.3. Nonlocal variable profile over the length approaches convergence with mesh refinement. Finer meshes show a slight increase in strains over the coarser mesh (fig. 3-13). This is a consequence of the evolving internal length which shrinks the influence domain when the element is completely damaged. Maximum values of ε eq are lower than that of κ due to the increased internal length and is similar to the observation made in section 3.4.

36 2 DAMAGE MODELS 3.6 NL VS SBNL DAMAGE MODEL Both standard and stress based nonlocal damage model solve the spurious mesh dependency observed in the local damage model as seen in sections 3.4 and 3.5. Stress based nonlocal model also rectifies the excessive damage diffusion seen in the standard nonlocal model as is observed from the narrow damage profiles shown in fig This model rectifies the diffusion of damage by using the concept of evolving internal length which shrinks the influence domain when maximum damage level is reached. 1.5 NL, ε eq, 15 mm l int NL, κ, 1 mm l int SBNL, ε eq, 15 mm l int SBNL, κ, 1 mm l int (a) Damage profile for case NL, ε eq, 15 NL, κ, 1 SBNL, ε eq, 15 SBNL, κ, (b) Damage profile for case 2. Figure 3-14: SBNL vs NL damage model..7.5 κ -.35 ε -.25 NL SBNL NL SBNL (a) κ profile for case 2. (b) ε eq profile for case 2. Figure 3-15: SBNL vs NL damage model. It is also observed that ε eq (NLV) with 15 mm internal length produces identical damage plateau near peak damage levels when compared to κ (NLV) with 1 mm internal length.

37 3.6 NL VS SBNL DAMAGE MODEL 21 The only drawback is that it tends to overestimate the spread beyond the peak. This is deemed acceptable as the increase is restricted to less than 8%, assuming the one predicted by κ is used as the benchmark. Another observation made is that the nonlocal strain variable profile showed higher values for stress based nonlocal model as shown in fig. 3-15a. This strong localisation is due to the evolving internal length concept which shrinks the influence domain at a completely damaged state.

38 22 DAMAGE MODELS 3.7 SUMMARY This chapter presented the results of the three models used in this study to predict tensile failure. These results serve as a benchmark for implementing numerical erosion. The major observations from this chapter are: The local damage model is mesh dependent and produced several localised peaks with mesh refinement which are a consequence of the explicit computation scheme used in LS-DYNA. The standard nonlocal model served as an improvement over the local model as the results are mesh independent irrespective of the choice of the nonlocal variable. In the standard nonlocal model, κ (nonlocal variable) produced excessive damage diffusion than ε eq (nonlocal variable) with the same value of internal length. This resulted in variation of the width of the FPZ predicted by the two variables. This indicated that the internal length is not purely a material property. It also depends on the choice of the nonlocal variable. To compensate for the variation in the damage profiles predicted by both nonlocal variables, it was decided to use 1 mm and 15 mm as internal length for κ and ε eq respectively. This ensured that the damage profile produced similar results at peak level. Numerical stability for the cases studied in this work is ensured with ρ min =.3 in the stress based nonlocal model. The stress based nonlocal model produced the most numerically reliable results when compared to the standard nonlocal and local models. It not only rectified the spurious mesh dependency of the local model but also solves the diffusion of damage seen in standard nonlocal model. In the stress based nonlocal model, the width of the FPZ predicted is identical for both nonlocal variables. This is due to the introduction of the concept of evolving internal length. It is however noted that the damage profiles are not identical, κ as the nonlocal variable results a wider profile, when compared to ε eq, due to its monotonically increasing nature. In order to obtain identical profiles at peak damage, the same values for internal length set in the standard nonlocal model is utilised. ε eq as the nonlocal variable alleviated the localised peaks that were obtained with κ as the nonlocal variable in both standard and stress based nonlocal model.

39 Chapter 4 EROSION IN LOCAL DAMAGE MODEL This chapter deals with assessing numerical erosion with the local damage model. As seen in section 2.3, there are stress, strain, and damage based erosion limits pre-defined in LS- DYNA. Out of these, the stress and strain limits work with any material model while the damage based limit works primarily with certain predefined material models. Since the influence of retaining the mass post-erosion is out of the scope of this work, the default option in LS-DYNA is used which discards the mass and energy upon deletion of elements. 4.1 STRESS BASED EROSION LIMIT LS-DYNA offers different types of stress based limits. Maximum principal stress, normal stress, and minimum principal stress are some examples of this feature. A detailed list on these can be found in the LS-DYNA keyword manual [15]. σ P1 P2 Strain Hardening Elastic Plastic Strain Softening Brittle Figure 4-1: Post-peak behaviours observed in material. Considering the different stress-strain curves observed in materials (fig. 4-1), a limiting value at position 1 (P1) can accurately represent materials exhibiting hardening. When this limit is lowered to position 2 (P2), it represents brittle behaviour but fails to represent both elastic-plastic and strain softening behaviour. ε

40 24 EROSION IN LOCAL DAMAGE MODEL In the local damage model, the extent of localisation depends on the mesh discretisation. Finer the mesh, larger is the extent of localisation observed. To illustrate this, it is decided to use finer meshes when case 2 (spalling failure) is assessed with numerical erosion. For a stress based erosion limit, a cut-off value is required. In case of tensile failure, the only logical limiting value is the tensile strength of the material under study which is 3. MPa (P2) as shown in fig The number of elements deleted for both case 1 and 2 (cantilever beam and spalling failure respectively) are shown in table 4-1. It is observed during the simulation that there is no damage propagation in both cases because the stress based limit fails to represent the softening behaviour. It is also observed that the deletion is irregular and mesh dependent (fig. 4-2). This is due to the mesh dependent localisations observed when local damage model is used to predict failure in chapter 3..1 m.1 m 5 Elements 75 Elements 1 Elements 25 Elements 1 Elements 5 Elements ω (a) Case 1 (cantilever beam)...5 ω 1. (b) Case 2 (spalling failure). Figure 4-2: Erosion with limiting σ max = 3. MPa. Table 4-1: Element deletion in case 1 and 2 σ max Erosion Limit 3 MPa Number of Elements Deleted Case 1 Case 2 5 (.1 m) 15 (.15 m) 6 (.8 m) 26 (.14 m) 1 (.1 m) 5 (.1 m) This deletion leads to a purely brittle failure and any limit lower than the tensile strength would delete the elements in the elastic domain. If a higher limit is prescribed, there is no deletion. It is therefore concluded that stress based erosion limits are unsuitable for the examples studied in this work.

41 4.2 STRAIN BASED EROSION LIMIT STRAIN BASED EROSION LIMIT In this section, maximum principal strain (ε max ) is used as an erosion limiter. Other available options are volumetric strain and maximum effective strain. Since the simulation is in a one dimensional premise (ν = ), identical erosion patterns are observed with varying parameters and are not shown in this report. L1 L2 L3 Figure 4-3: Strain values used for erosion limit. Three points of interest are picked from the stress-strain curve. These correspond to a strain level of ε max = at tensile limit of 3. MPa (L1), ε max = (L2) at a point where the material has lost some stiffness, and ε max = (L3) at failure. These points are indicated by red circles in fig At a limit of ε max = 1 1 4, there was no damage in the material when elements are deleted. As explained earlier, the deletion represents brittle failure. Element deletion is also irregular, mesh dependent and identical to results shown in fig At a limit of ε max = 3 1 4, there is evolution of damage before the elements are deleted. This is expected as the point is positioned in the descending branch of the stress-strain curve. Peak-damage observed before deletion is at ω =.7 and the deletion is irregular as shown in fig. 4-4 and 4-5. At a limit of ε max = 3 1 3, the damage level reached ω = 1., but the deletion is still irregular. This is not shown as it exhibits identical behaviour to the case with ε max = Irrespective of the limiting value, numerical erosion resulted in highly irregular and mesh dependent deletion. Therefore, it is concluded that mesh dependent erosion is a consequence of the mesh dependent results obtained from the local damage model (as seen in chapter 3).

42 26 EROSION IN LOCAL DAMAGE MODEL.1 m 5 Elements 75 Elements 1 Elements..5 ω.7 Figure 4-4: Erosion in case 1 with limiting ε max = m 1 Elements 25 Elements 5 Elements..5 ω Figure 4-5: Erosion in case 2 with limiting ε max =

43 4.3 DAMAGE BASED EROSION LIMIT DAMAGE BASED EROSION LIMIT A user-defined algorithm is implemented in the code to erode elements using damage as the erosion limiter. Limiting value of ω =.999 is prescribed for both cases as it is logical to erode the element when failure has occurred (zero stiffness). Having seen extremely localised failure patterns in fig. 3-5, it is expected that this limit will delete elements which are mesh dependent and irregular. It is seen from fig. 4-6, that this is indeed the case. The elements deleted are the ones over which damage has localised in chapter 3. It is also observed that the results with maximum principal strain limit of showed identical results as with the damage limit of.999. This is expected, as both limits are at failure, where damage has peaked and the element has zero stiffness..1 m.1 m 5 Elements 75 Elements 1 Elements 25 Elements 1 Elements 5 Elements ω (a) Case 1 (cantilever beam)...5 ω 1. (b) Case 2 (spalling failure). Figure 4-6: Erosion with limiting ω =.999. Although the deletion observed is mesh dependent, it is important to note the ease with which the limiting value can be formulated for damage based erosion limits. Unlike the strain based limits, damage based erosion does not need in-depth understanding of the constitutive law or a numerical assessment for formulating the limiting value.

44 28 EROSION IN LOCAL DAMAGE MODEL 4.4 SUMMARY Erosion in local damage models produced the following conclusions: Stress based erosion limits deleted elements in the elastic domain and can represent pure brittle failure. They cannot represent the exponential softening law describing the material and are unsuitable for the cases studied in this work. Both strain and damage based limits produced mesh dependent deletion. This is because of the spurious mesh dependency suffered by the local damage model. Formulation of limiting values for strain based erosion required an in-depth understanding of the material model. Formulation of limiting values for damage based erosion is straightforward and does not need in-depth understanding of the material model. These results show irregular deletion pattern due to the large extent of localisation because of the usage of an explicit computation scheme. The classical solution [14], where an implicit computation scheme is used, would have restricted the localisation to one element and the erosion algorithm would have deleted that element.

45 Chapter 5 EROSION IN NONLOCAL MODELS In chapter 3, two nonlocal models are used to predict failure, namely, the standard nonlocal (NL) and the stress based nonlocal (SBNL) damage models. In this chapter, numerical erosion is implemented in these models with an objective to assess the different limits for their robustness and mesh sensitivity. Upon completion, it is recommended to add the best performing limits with the models to further improve them. 5.1 NONLOCAL COMPUTATION SCHEME A brief introduction of the computational sequence used is shown in fig LS-DYNA uses an explicit computation scheme. Initially, the increment in the global strain tensor is calculated and stored. Since a user defined material model is used, after the calculation of the global strains, the code enters the part describing the user defined material (UMAT). Inside the UMAT, equivalent strain is first calculated based on the global strain using the modified von Mises equivalent strain described in chapter 2. From the equivalent strain, the history variable κ is calculated. Based on the choice of the nonlocal variable, the code replaces the local variable with its nonlocal counterpart. The nonlocality is introduced by means of an internal length, over which weighted average of the strains are made (using Gauss distribution). After this, κ is updated to store the historical maximum of the nonlocal variable which is used to calculate damage. Once the damage variable is obtained, the stress tensor is updated and the time step is complete. The data stored in this time step is then used to compute the global strain increment in the next time step and the process repeats till the end simulation time is reached. The stability of an explicit scheme depends on the scaling factor used on the critical time step (t crit ) obtained using Courant condition. In this work, the critical time is scaled down to.1 t crit to ensure stability.

46 3 EROSION IN NONLOCAL MODELS Global strain increment ( ε) Enters UMAT Calculation of ε eq Calculation of κ ε eq Based on NLV Update κ κ Time step increment ( t) ω Calculation of stress (σ) Figure 5-1: Computation scheme used in the code. 5.2 EROSION IN STANDARD NONLOCAL MODEL The possibility of utilising erosion limits tested in the previous chapter (strain ε max and damage ω based) are analysed for the standard nonlocal model. In addition to the existing limits, internal strain variables ( κ and ε eq ) are also explored Strain based erosion limit Using the numerical assessment made in the chapter 4, two limits of and were used for principal strain in both cases is excluded as a limit in this chapter as it is already established that the elements deleted forces brittle failure, rather than strain softening. The initial simulations use κ as the nonlocal variable. For case 1 (cantilever beam), the first erosion limit of ε max = resulted in mesh dependent and irregular element deletion as shown in fig. 5-2a. Similar results are obtained for case 2 (spalling failure) and is shown in fig. 5-2b. This irregular deletion is due to the mesh sensitive nature of global strain variable. In an explicit computation scheme, the strains are calculated based on the data from the previous time step. It is expected to have small fluctuations in the strain

47 5.2 EROSION IN STANDARD NONLOCAL MODEL 31 as no local convergence is guaranteed with an explicit scheme. Therefore, small deviations from the exact solution is expected without compromising the global response. Deletion of elements before the code enters UMAT (fig 5-1) disturbs the nonlocal averaging which leads to further fluctuations in the global strains..1 m.1 m 1 Elements 25 Elements 1 Elements 25 Elements 5 Elements 5 Elements..4.8 ω (a) Elements deleted in case ω.8 (b) Elements deleted in case 2. Figure 5-2: Results with limiting ε max =3 1 4 (κ NLV). Using limiting ε max = 3 1 3, resulted in the deletion of the first element in case 1, thus exhibiting mesh dependent (MD2) deletion as shown in fig Elements deleted correspond to a zone of.1 m,.4 m and.2 m for 1, 25 and 5 elements mesh respectively. 1 Elements 25 Elements.1 m El 25 El 5 El 5 Elements..5 ω 1. (a) Elements deleted over the length (b) Damage over the length. Figure 5-3: Erosion in case 1 (κ NLV) with limiting ε max =3 1 3.

48 32 EROSION IN NONLOCAL MODELS In case 2, the deletion is highly irregular and mesh dependent (MD2) as seen in fig Elements deleted and their corresponding zone for case 2 is shown in table 5-1. The irregular deletion is attributed to the global nature of the principal strain variable. The mesh sensitive nature is a consequence of the usage of an explicit computation sequence explained earlier. The peak damage value also varied (reaching ω =.7 for ε max = and ω = 1. for ε max = ). This is in concurrence with the observation made in chapter 4. 1 Elements 25 Elements 5 Elements.1 m..5 ω 1. (a) Elements deleted over the length El 25 El 5 El (b) Damage over the length. Figure 5-4: Erosion in case 2 (κ NLV) with limiting ε max = Table 5-1: Element deletion in case 2 (κ NLV). ε max Erosion Limit Mesh Size Elements deleted (.76 m) 1 9 (.9 m) 5 35 (.7 m) The deletion is also influenced by the erosion limit variable. Principal strain being a type A strain variable, drops to zero upon the deletion of the first element as shown in fig In case 1, this causes the deletion to stop after one element is eroded. It is also the reason for the decrease in the width of the deleted zone in case 2. Using ε eq as the nonlocal variable produced similar results (mesh dependent, MD2) and is not presented in this report. From this section, it is concluded that global strain variables available in LS-DYNA cannot facilitate mesh independent erosion in standard nonlocal model due to their mesh sensitive nature which is a consequence of the explicit computation scheme used in LS-DYNA. The introduction of nonlocality alleviates the extent of the mesh dependence as the global strains are calculated after the nonlocal averaging in the computation scheme (fig. 5-1)

49 5.2 EROSION IN STANDARD NONLOCAL MODEL 33 εmax.3.15 Element 1 Element 2 Element 3 Element 4 5x1-5.1 Time (s) Figure 5-5: Evolution of principal strain (type A) post-erosion in case Nonlocal variable based erosion limits The nonlocal variable κ is used as an erosion limit to assess if it performs better than principal strain. Both and are used as limiting values. Using a limit of κ = (κ as the nonlocal variable) resulted in mesh independent element deletion (MI1) for both cases as shown in table 5-2. However, the deletion with the same limit for the ε eq (ε eq as the nonlocal variable) is mesh dependent (MD2). Only one element is deleted for both cases. This is due to ε eq being a type A strain variable and once deleted at a lower level, drops to zero (unloading process). The damage levels peaked at.7 to.78 with this erosion limit and is in concurrence with the observation made in the previous sections. Table 5-2: Element deletion with κ as the NLV. κ Erosion Limit Mesh Size Elements Deleted Case 1 Case (.12 m) 27 (.18 m) 1 1 (.1 m) 1 (.1 m) 5 6 (.12 m) 59 (.118 m) The second erosion limit of κ = resulted in mesh independent deletion (MI1) for both cases as shown in fig. 5-6 and 5-7. The nonlocal formulation eliminates the mesh sensitive nature of the strain variable by replacing it with its mesh independent nonlocal counterpart. The elements deleted are shown in table 5-3. It is influenced by the type of erosion limit used. As κ is a type B strain variable, it cannot decrease. Therefore, results in deletion of elements in successive time steps.

50 34 EROSION IN NONLOCAL MODELS 1 Elements 25 Elements.1 m El 25 El 5 El 5 Elements..5 ω 1. (a) Elements deleted over the length (b) Damage over the length. Figure 5-6: Erosion in case 1 (κ NLV) with limiting κ= Elements 25 Elements 5 Elements.1 m..5 ω 1. (a) Elements deleted over the length El 25 El 5 El (b) Damage over the length. Figure 5-7: Erosion in case 2 (κ NLV) with limiting κ= Table 5-3: Element deletion with κ as the NLV. κ Erosion Limit Mesh Size Elements Deleted Case 1 Case (.12 m) 26 (.14 m) 1 1 (.1 m) 1 (.1 m) 5 6 (.12 m) 51 (.12 m) The second erosion limit of ε eq = resulted in mesh dependent deletion in both cases (table 5-4). For case 1, only the first element is deleted (MD2) which is identical to the deletion observed with the same limiting value for principal strain (fig. 5-3). The deletion

51 5.2 EROSION IN STANDARD NONLOCAL MODEL 35 is again influenced by the type of erosion limit. As ε eq is categorised under type A strain variable, it drops to zero and prevents further deletion. In case 2, the deletion is mesh dependent (MD2). However, irregular deletion is not observed (fig. 5-8). Deletion using these limits are after the code has entered UMAT (fig. 5-1) and the nonlocal averaging eliminates the mesh sensitivity which was observed when global strain based limit was assessed. The difference in deletion between the two variables is due to the categorisation of the variables ( ε eq is type A strain variable while κ is type B strain variable ). 1 Elements 25 Elements 5 Elements.1 m..5 ω 1. (a) Elements deleted over the length El 25 El 5 El (b) Damage over the length. Figure 5-8: Erosion in case 2 (ε eq NLV) with limiting ε eq = Table 5-4: Element deletion with ε eq as the NLV. ε eq Erosion Limit Mesh Size Elements Deleted Case 1 Case (.4 m) 17 (.68 m) 1 1 (.1 m) 8 (.8 m) 5 1 (.2 m) 28 (.56 m) Note:- There is a possibility of using κ as the nonlocal variable with ε eq as the erosion limit. This results in irregular deletion similar to the cases when the principal strain is used as the erosion limit. The reason for this is that the ε eq is a local variable and is mesh sensitive. Unless its nonlocal counterpart is used, mesh dependency prevails. There is also a possibility of using ε eq as the nonlocal variable with κ as the erosion limit. This result is identical to the ones shown in table 5-4. This is because κ is programmed to store the historical maximum of the nonlocal counterpart of ε eq in the computational sequence (fig. 5-1). Since ε eq is a type A variable, it drops to zero upon deletion. Therefore, it can cause no further increase in κ. These observations indicate that the choice of the nonlocal variable also influences the deletion.

52 36 EROSION IN NONLOCAL MODELS Damage based erosion limit Damage is calculated from the nonlocal variable used for the simulation (fig. 5-1). Using damage as an erosion limiter has several advantages. Firstly, it is an indirect representation of nonlocal strain. Secondly, it represents the degradation in stiffness which is used to calculate the stress as shown in section A limit of.999 is used for both cases as it directly removes the completely damaged (zero stiffness) elements. Damage profiles post-erosion using damage based erosion limit for case 1 is shown in fig. 5-9 with the number of elements deleted summarised in table 5-5. Using ε eq as the nonlocal variable resulted in deletion of the first element only, thus exhibiting mesh dependent deletion (MD2) as seen in fig. 5-9a while κ as the nonlocal variable resulted in mesh independent deletion (MI1) as seen in fig. 5-9b El 25 El 5 El El 25 El 5 El (a) Damage over the length (ε eq NLV) (b) Damage over the length (κ NLV). Figure 5-9: Erosion in case 1 with limiting ω=.999. Table 5-5: Element deletion in case 1. ω Erosion Limit.999 Mesh Size Elements Deleted ε eq NLV κ NLV 1 1 (.1 m) 1 (.1 m) 25 1 (.4 m) 3 (.12 m) 5 1 (.2 m) 6 (.12 m) In case 2, using ε eq as the nonlocal variable with damage based erosion limit resulted in mesh dependent deletion approaching mesh independence (MD1) with mesh refinement as seen in fig. 5-1a. This result shows a marginal improvement over the erosion limit with ε eq = seen in the previous section. Using κ as the nonlocal variable with damage based limit resulted in mesh independent deletion (MI1) as seen in fig. 5-1b. This is also identical to the results obtained with κ = (κ as the nonlocal variable).

53 5.2 EROSION IN STANDARD NONLOCAL MODEL El 25 El 5 El El 25 El 5 El (a) Damage over the length (ε eq NLV). (b) Damage over the length (κ NLV). Figure 5-1: Erosion in case 2 with limiting ω=.999. Table 5-6: Element deletion in case 2. ω Erosion Limit.999 Mesh Size Elements Deleted ε eq NLV κ NLV 1 6 (.6 m) 1 (.1 m) (.52 m) 25 (.1 m) 5 23 (.46 m) 5 (.1 m) The difference in deletion seen in table 5-6 is attributed to the type of nonlocal variable chosen for the simulation. Until this section, it is observed that the element deletion is influenced by the type of variable chosen as the erosion limiter. In this section, the choice of nonlocal variable is influencing the elements deleted as damage is obtained from the nonlocal variable used. While damage is a monotonically increasing variable similar to type B variable, the nonlocal variable used to compute damage is the deciding factor. In case 2, ε eq (type A, nonlocal variable) with damage as the erosion limit deleted less elements than κ (type B, nonlocal variable). This behaviour is similar to the one obtained in case 1, but the number of elements deleted is more than one in all meshes studied. A possible explanation to this is that in case 2, there is a large number of elements that reach the damage limit simultaneously. From the number of elements deleted, it is clear that damage based limits produced comparable results as the nonlocal strain based limits. There is also no requirement of any calibration as discussed in chapter 4. A limit close to 1. can delete elements that are completely damaged irrespective of the case. If the material model can produce accurate results with damage as an available variable. It can be used as an erosion limit and offers the best chances of obtaining mesh independent element deletion for the standard nonlocal model. The added advantage using damage based limit is that the limiting value can be formulated without requiring in-depth knowledge of the constitutive law used.

54 38 EROSION IN NONLOCAL MODELS 5.3 EROSION IN STRESS BASED NONLOCAL MODEL This section is built up in a similar sequence as section 5.2. Principal strain (ε max ) and internal variables ( κ or ε eq and ω) are evaluated as the erosion limits Strain based erosion limit The first limit (ε max = ) produced results similar to the ones seen in section Since the deletion is irregular, it did not provide any extra information and is not presented in this report. With both κ and ε eq as the nonlocal variable, the second erosion limit of ε max = resulted in mesh independent deletion (MI1) for case 1 as seen in table 5-7. Damage profiles post-erosion with κ as the nonlocal variable is shown in fig. 5-11b. 1 Elements 25 Elements.1 m El 25 El 5 El 5 Elements..5 ω 1. (a) Elements deleted over the length (b) Damage over the length. Figure 5-11: Erosion in case 1 (κ NLV) with limiting ε max = Table 5-7: Element deletion in case 1 for both nonlocal variables. ε max Erosion Limit Mesh Size Elements deleted (.12 m) 1 1 (.1 m) 5 5 (.1 m) For case 2, ε eq as the nonlocal variable resulted in mesh dependent deletion approaching mesh independence (MD1) with mesh refinement (table 5-8) due to its type A nature while κ as the nonlocal variable resulted in mesh independent deletion with minor irregularities (MI3). Only the results for erosion with κ as the nonlocal variable is shown in fig