Development and Application of MultiâScale Numerical Tool to ...

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DOI: 10.1002/adem.201400157

Development and Application of Multi-Scale Numerical Tool to Modeling Pneumatic Ventricular Assist Devices with Increased Athrombogenicity** By Magdalena Kopernik

The multi-scale numerical model of blood chamber of ventricular assist device (VAD) with deposited a biocompatible titanium nitride (TiN) nano-coating is introduced. The model is developed to optimize shape parameters of the blood chamber and to determine fracture parameters of the nano-coating. The development of finite element (FE) macro-model is combined with a digital image correlation data to validate a computed strain state. Contrary, on a micro-scale calibration and validation of a representative volume element (RVE) model of the wall of VAD composed of the TiN coating and substrate-polymer is realized by comparison with results of an experimental in situ SEM’s micro-tensile test. The multi-scale model of VAD is enriched with materials research. Thus, tensile tests of VADs’ polymers are performed to obtain the properties of designed materials of the VADs. On the other hand, on a micro-scale, the properties of TiN are identified based on a nanoindentation test and an inverse analysis for its interpretation. The profilometric studies and analytical model are presented to calculate a residual stress in the TiN. The final result of the paper is the multi-scale numerical tool to modeling pneumatic VADs with increased biocompatibility produced in Poland. 1. Introduction A ventricular assist device, or VAD, is a mechanical circulatory device that is used to partially or completely replace the function of a failing heart. Some VADs are intended for short-term use, typically for patients recovering from heart attacks or heart surgery, while others are intended for long-term use (months to years and in some cases for life), typically for [*] Dr. M. Kopernik AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Kraków, Poland E-mail: [email protected] [**] Financial assistance of the National Science Center in Poland, project no. 2011/01/D/ST8/04087, is acknowledged. The experiments were performed at the WIMiIPAGH University of Science and Technology in Cracow in Poland and at the Foundation of Cardiac Surgery Development in Zabrze in Poland. 278

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patients suffering from congestive heart failure.[1] VADs are designed to completely take over cardiac function and generally require the removal of the patient’s heart.[2] They are used to assist either the right (RVAD) or left (LVAD) ventricle, or both at once (BiVAD). Which of these types is applied depends primarily on the underlying heart disease and the pulmonary arterial resistance that determines the load on the right ventricle. LVADs are most commonly used, but when pulmonary arterial resistance is high, right ventricular assistance becomes necessary. Long-term VADs are normally used to keep patients alive with a good quality of life while they wait for a heart transplantation, which is known as a “bridge to transplantation.” However, LVADs are sometimes used as destination therapy and sometimes as a bridge to recovery. In the last few years, VADs have improved significantly in terms of providing survival and quality of life among recipients.[1] Polyurethane is used as a designed material of the VAD. The polymer exhibits the change in stiffness in the long-term

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ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

M. Kopernik/Development and Application of Multi-Scale Numerical Tool …

1) modeling of prototype focused on optimization of selected shape and material parameters, as well as its impact on flow and behavior of the blood, 2) development of robust blood rheology model, 3) modeling problems concerning the interaction of biological and artificial materials, 4) creation of mathematical models of the experimental nanotests for nano-coatings, which build wall of the VAD and the identification of mechanical properties for nano-materials. The contractors of the described task decided to develop own software, because it is not possible to develop complete numerical model of VAD based on commercial software. Using commercial software, the one model can be selected and then, modified by applying available tools. The development of own code enables its future extension and its cooperation with different modeling modules on microand nano-scales. The full flexibility in adaptation of the program to future requirements is possible, if only the access to source code is opened and the code is well known. The last fact only exists while development of a proprietary software. The advantages of FEMVAD program are as following: 1) Automatic method of removing the non-compliances in the CAD models of VADs. Magdalena Kopernik graduated with BSc in medical physics in 2002 and MSc in experimental physics in 2004 at the University of Silesia in Poland. She received her PhD in 2008 from the Silesian University of Technology. She has worked at the AGH University of Science and Technology since 2006 and now she is an assistant professor. She has collaborated with the Foundation of Cardiac Surgery Development since 2002 and focused her research on solving problems related to numerical and experimental aspects of materials and designs applied in the Polish ventricular assist devices.


2) Automatic fragmentation of FEM mesh generated in any FE program on a macro-scale to set precisely loadings and boundary conditions in any place of macro-model of VAD’s blood chamber. 3) Introduction of multi-scale approach, which allows the analysis of local state of stress and strain, what is a valuable possibility, because nano-coatings will be applied as designed materials in VADs. The practical aspect of the multi-scale model of mechanical phenomena in the blood chamber of VAD is based on comparative analysis of states of stresses and strains for different shapes of VADs, optimization of parameters of nano-coatings, determination the regions of surface of VAD, for which the loss of cohesion for the material system coating/substrate is probable. 4) Identification of parameters in material model of TiN by development of nano-model and program to simulation the nanoindentation test for TiN/PU with the option of inverse analysis and modeling the unloading process. 5) Fast optimization. The short computation time needed for a 3D nonlinear solution, the completeness of the multiscale model of the VAD, implemented optimization algorithms enable to optimize, for example, the shape parameters and any parameters of any versions of models of the VADs in the VADFEM computer program. 6) The possibility of control the numerical errors and their reduction by applying the procedures of mesh adaptation (adaptation directed on the specified goal – goal-oriented adaptation). 7) The possibility of reaching very precise results, because of conduction the simulation on parallel computing machines. 8) The possibility of saving results in any defined formats, free transfer of files among particular modules of the program, importation of meshes generated in own FE code, importation of meshes generated in any FE codes and possibility of FE mesh processing and adaptation using developed algorithms. The advantages of own codes related to fluid dynamics are not presented, because author of the paper did not participate in their implementation and development. The FE fluid solution communicates with the FE solid solution by the distributions of blood pressures. The latest pneumatic Polish left VAD (Religa Heart Ext, Figure 1) is composed of a blood chamber and two connectors.[4] The assembled device is ca. 157 mm in length, 56 mm in width, and 51 mm in height. The basic hydrodynamic parameters of VAD are as follows: an average flow of 2–5 dm3 min1, a ventricular stroke volume of 80–90 cm3, a typical loading pressure of 90–160 mmHg in the outlet connector and 15 mmHg in the inlet connector and a pressure rise of less than 4700 mmHg s1. The design of the proposed Polish LVAD is closest to the Medos HIA-VAD (Germany) and is one of the designs of medical pumps developed in the Polish Artificial Heart Programme.[5] Similar pneumatic and polymeric VADs include the following models: Abiomed AB5000 (USA), ThoratecPVAD (USA), and Berlin Heart EXCOR (Germany).[6,7]


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clinical application. It was found that the blood clot formation proceeds in the polyurethane/blood contact area and the polyurethane is subjected to degradation. Therefore, there is a necessity to modify the surface of polyurethane by covering it with an athrombogenic coating. Biomaterials such as: titanium Ti, and stoichiometric titanium nitride (TiN) as well as titanium carbo-nitride Ti(C,N), seem to be good candidates for blood-contact applications such as VADs.[3] The author of the paper participated in the “Polish Artificial Heart Programme” and realized the solid model in task: “Numerical model of totally implantable pulsate VAD.” The goal of the task was development of numerical model of pulsatile VAD and computer program with the option of a local multi-scale modeling, which allows the analysis of the problems posed by the VAD’s application in terms of blood flow through prosthesis. The main defined problems included:

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Fig. 1. A CAD model (a) and a physical model (b) of the VAD: 1 and 2–connectors; 3 and 4–valves; 5–a blood chamber; 6–a pneumatic chamber with membrane (not clearly visible in the selected photo). Reproduced with permission.[33] Copyright 2014, Techno Press.

However, the proposed Polish LVAD is a unique design, because the blood chamber has a layered structure to reduce probability of thrombus formation and to improve hemocompatibility.[8,9] Applying the biocompatible TiN nanocoating on surface of the Religa Heart Ext can lead to a loss of cohesion of connection nano-coating/substrate in a cyclic working conditions of the blood chamber. The possibility of the last disadvantage causes necessity of advanced experimental and numerical studies dedicated to materials and design of the Religa Heart Ext. Thus, the authors’ present work can be considered state of art. The goal of the paper is development and application of numerical tool to multi-scale modeling of Polish prototypes of pneumatic VADs with increased athrombogenicity considering detailed materials research.[10–13] The specific objectives of the work are formulated as following: a) The objectives of FE model on a macro-scale are: 1) FE mesh processing.[14] 2) Solution of a problem of convergence in a nonlinear tasks (a nonlinear elasticity and an elasticplasticity).[11] 3) Numerical validation of the developed FE code.[15] 4) Comparison of the FE models of prototypes of VADs.[15] 5) Experimental validation of the developed FE code.[16] 6) Identification of properties of polymers applied in the VADs.[10,13,16] 7) Optimization of the model of prototypes of VAD.[17] b) The objectives of FE model on a micro-scale are: 1) Identification of mechanical properties of the TiN nano-coating.[12,18,19] 2) Analysis of a strain and a stress states in a multi-layer wall of the prototypes of VADs.[20] 3) Sensitivity analysis of the developed micro-model of mechanical test.[21,22] 4) Identification of fracture parameters of the TiN nanocoating.[13,23] 280


2. FE Model on Macro-Scale 2.1. Description of Multi-Scale Model The realization of specified goals on macro- and micro-scales leads to development of the VADFEM computer program, which allows to model the strain–stress states in the multi-layer prototypes of VADs. The multi-scale model of VAD developed in the FE code is composed of a macro-model of the blood chamber and a micro-model of the VAD’s wall composed of the TiN nano-coating deposited on polymer. The micro-scale problem is solved for a representative volume element (RVE). The considered multi-scale model is realized in two stages (Figure 2): a residual stress modeling and a modeling under work load.[11] In the first stage (Figure 2a), the residual stress is reached by applying experimental results on a micro-scale. In the second step (Figure 2b), the stress and strain states computed in the macro-model under static homogenous distribution of blood pressure (set on the inner surface of blood chamber) located in the failure source areas between two connectors of VAD are used as initial values in the micromodel. In this stage, boundary conditions are moved from solution on a macro-scale to a micro-scale. The suggested multi-scale approach is classified as an upscaling method based on the FE solutions in two different scales for analysis the phenomena occurring in the nanocoatings due to load of the VAD on a macro-scale.[24] Similar solution based on the FE method combined with the RVE approach is proposed in literature.[25] 2.2. The Bases of FE Model on Macro-Scale The implementation of FEM shown in literature is dedicated to modeling of the processes observed in VADs and human heart, for example: a blood flow in different phases of work of VAD,[26] modeling of a human heart valve function and a multiscale modeling of a fiber structure of a human heart.[27] The base conception of FEM for elastic and elastic–plastic problems is also widely described in many works.[28] The representative example of CAD’s models of prototypes of the Polish VADs investigated in the present paper (Figure 1a) is composed of two main parts: the top pneumatic chamber with membrane and the bottom blood chamber, as well as two channels: reduced connectors with valves.[5]




KM ¼

Z ½BT ½DM ½BdV

ð2Þ

Z FM ¼ ½NT pM dS

ð3Þ

V

S

where S is a contact surface, pM is the pressure inside the blood chamber of the Polish VAD, [B] is the matrix containing derivatives of shape functions, [DM] is the matrix containing the appropriate material properties (EM, nM), [N] is the matrix of shape functions of a finite element (FE), V is the volume. A tetrahedron FE with a five-point scheme of integration is used in the macro-model of VAD. The average number of applied nodes is 50 000, and the average number of applied tetrahedron FEs is 150 000. The boundary conditions applied in the macro-model of VAD are in accordance with settings of the physical experiment (Figure 3) and they are Fig. 2. The general conception of the multiscale model of VAD's blood chamber: (a) a residual stress as follows: (i) distribution of a blood pressure modeling and (b) an active load modeling. Reproduced with permission.[15] Copyright 2011, Wroclaw University of Technology. pM on an inner surface of the blood chamber, (ii) fixed surfaces in an outer upper part of the The macro-model of the Polish VAD is prepared in the blood chamber (no displacement in Z direction), and (iii) not VADFEM computer program in which a thermo-mechanical fixed surfaces in an outer part of the blood chamber (free task is solved. The FEM macro-model of the VAD allows to surfaces).[16] distinguish a failure source areas of the loaded model on the 2.3. The FE Mesh Processing basis of a strain and a stress analysis. The macro-scale boundary problem is formulated by the theory of a nonlinear elasticity The main requirements for VADFEM computer program and a distribution of displacements UM i . The proposed are to load and preprocess FE meshes for any software. approach describes deformation of the blood chamber of Therefore, it is necessary to develop algorithms of separating VAD under a blood pressure, if: a stress is related to a strain by a from the whole FE mesh of the surface of the mesh elements nonlinear equations (according to the nonlinear theory of that come into contact only with the surroundings. It is elasticity) and a strain disappears in an unloading conditions. prepared based on the fact that each mesh contains data about The nonlinearity in an elastic deformation process of the blood the coordinates of nodes and their connections in the FEs. chamber is a result of: a nonlinear mechanical properties of SłuchajIt should be noted that in the majority of the known the polymer (Chronothane 55D, ChronoFlex C 55D and/or commercial programs, FEM does not provide information on Bionate II 55D; compare Section 2.7) and a nonlinear the mesh, which allows to distinguish whether the node is an mechanical properties of the TiN nano-coating.[10,12,13,16] external or internal (lack of documentation to the generated The problem of the blood chamber deformation on a macrofiles, which contain detailed information about the mesh). scale is considered as a 3D solution. Thus, the defined boundary Methodology for determining nodes on the surface and problem of a theory of nonlinear elasticity is composed of the surfaces is implemented through a basic algorithm described groups of equations described in a literature.[29] The Young’s below: modulus is assumed as a constant value in a case of a small 1) Four surfaces are analyzed for each element of the FE mesh nonlinearity of material. In the nonlinear zone of deformation M (e.g., the FE mesh developed in the ABAQUS FE code). an effective stress s M is a function of a deformation e and a i i 2) In the rest of FE mesh a surface is searched, which contains temperature t. The effective modulus EM is used instead of the p the same nodes. If such a surface exists, the both surfaces Young’s modulus in each iteration: M are considered as internal boundaries of the area. However, s i EM ð1Þ p ¼ M in the opposite case, an outer surface is considered. e i




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M Therefore, a relation s M i ðei ; tÞ is obtained in a tensile test. The components of a stiffness matrix [K] and a complete load vector {F} are written according to the forms:

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M. Kopernik/Development and Application of Multi-Scale Numerical Tool … The possibility of automatic fragmentation of FE mesh generated in any program on a macro-scale is important to set precisely a load and boundary conditions in any place of the macro-model of VAD’s blood chamber. In the first stage of calculation, the boundary conditions are used according to Figure 3b and the result of a separation the external surface from the internal is visible in Figure 4. The separation of fixed nodes is carried out on a basis of analysis of a slope of surface. According to presented result, the proposed Fig. 3. The Polish VAD: (a) Settings of the physical model are marked on an image captured by one of the procedure and algorithm applied to separacameras and (b) Boundary conditions of the FE macromodel and coordinate system orientation (red – pressure, green – fixed surface, blue – free surface). Reproduced with permission.[16] Copyright 2012, tion the nodes of FE mesh allow setting the Wroclaw University of Technology. boundary conditions in any location of the analyzed blood chamber of VAD. The final results of mesh separation for two prototypes of the Polish 3) The solution assumes that all external nodes belong to the VADs are shown in Figure 4.[17] external surface, while the remaining nodes are internal. It is also developed a second method of separation the internal from the external surface, in which the cloud of points is generated around the FE mesh and around a center point of blood chamber. The center point inside the FE mesh is located exactly inside the blood chamber, and more specifically coincides with its center of gravity. Each node, which is the closest to the central point of blood chamber, is recognized as an internal node. This approach is proposed to verify additionally the eligibility of the node (external or internal). It is also decoupled from the FE type and specificity of the commercial FE program. The various steps of this algorithm lead to:[14] 1) Determination the distance of each node from a central point, defined as a center of gravity of the mass. 2) Separation of the inner surfaces of the all surfaces based on the directional cosine of the surfaces of elements. 3) Separation of the inner surfaces of the all surfaces based on the directional cosine of the surfaces of elements while reducing the distance to the closest surroundings of the central point. 4) Separation of the inner surfaces of the all surfaces based on the directional cosine of the surfaces of elements while reducing the distance to the closest surroundings of the central point and applying algorithm of current surface.

2.4. The Solution of Problem of Convergence in Nonlinear Tasks The problems of a convergence in a nonlinear task (a nonlinear elasticity and an elastic-plasticity) on a macro-scale are solved by applying a relaxation iterative method, which is described in details in literature.[30,31] The relaxation iterative method is adjusted to computing the effective Young’s modulus in the present work. In a standard iteration method, the effective Young’s modulus Ep is calculated in the next iteration by the formula as follows: ¼ Eðpþ1Þ p

ðpÞ

si

ðpÞ

ð4Þ

ei

is the effective Young’s modulus in the next where Eðpþ1Þ p ðpÞ ðpÞ iteration, s i and ei is an effective stress and an effective strain in a current iteration. According to the relaxation iterative method, the effective Young’ modulus in the next iteration is computed by the form as follows: ðpÞ ðpÞ Ep ¼ Ep þ v Epðpþ1Þ Ep ð5Þ where v is a relaxation coefficient (in the present work: v ¼ 1.0–1.6).

Fig. 4. The 3D FE models of two prototypes of Polish VADs after mesh separation: (a) the POLVAD and (b) the POLVAD_EXT. Reproduced with permission.[15] Copyright 2011, Wroclaw University of Technology.

282





Ep ¼ Eðpþ1Þ p

ð6Þ

The results of tests performed for the relaxation iterative method are shown in Figure 5.[11] The relation between computed value of an error of the effective Young’s modulus and a number of iterations for different values of a relaxation coefficient v is presented in Figure 5a. The error of the effective Young’s modulus d is calculated by applying the formula: d¼

ðpþ1Þ EðpÞ p Ep

ð7Þ

ðpÞ

Ep

where EðpÞ p is the effective Young’s modulus in the previous iteration, Eðpþ1Þ is a value calculated by using Equation 4. p The plotted error of the effective Young’s modulus d approaches to zero. In Figure 5b, the minimum number of iterations equal to 6 is reached for the relaxation coefficient v ¼ 1.2. The proposed iterative method is helpful to decrease twice a numberofiterationsand, therefore, a computing time required to obtain solution on a macro-scale is significantly reduced. In conclusion, a fast convergence in a nonlinear problems is particularly important in the macro-model examined in the present paper due to a big number of elements and a type of load-pressure. The similar nonlinear problems are also observed in the micro-model introduced in the present work, but a number of elements is much lower and a type of load– displacement does not enforce a big number of iterations.

The goals of simulations developed for VADs under operating conditions presented in the section are: (i) a verification of the results computed on a macro-scale in the VADFEM code by using a commercial FE code ABAQUS and (ii) a comparison between the models of prototypes of VADs: the POLVAD and the POLVAD_EXT in the VADFEM code. The results computed for VADs’ prototypes: the POLVAD and the POVAD_EXT in the VADFEM code and the ABAQUS code are shown in Figure 6 and in Table 1 as distributions of an effective strain, an effective stress, and a pressure under working conditions.[15,16] The biggest values of effective strain and effective stress in the wall of blood chamber are observed on the inner surface of bottom part of blood chamber between two connectors. The computed effective strains are in a linear elastic zones of values for polymer in both prototypes of the VADs. The results of simulations show values of effective stress equal tens of kPa on the wall of the blood chamber, what in comparison with applied load and material properties of the blood chamber can be assumed as correct.[10,12,13,16] The maximum values of all computed parameters shown in Table 1 and in Figure 6 are smaller for the POVAD_EXT. The good agreement between results computed in the commercial FE code and the VADFEM code is achieved for the model of POLVAD_EXT. The slightly worse agreement between results computed in the commercial code and the VADFEM code is achieved for the model of POLVAD, especially for effective stress values for which the difference is 17%. The disagreement can be caused by different method of solution the nonlinear task and different scheme of integration. The difference in results about over a dozen of percent is not a big difference in solution of nonlinear tasks. 2.6. Experimental Validation of Developed FE Code

2.5. Numerical Validation of Developed FE Code and Comparison of FE Models of Prototypes of Ventricular Assist Devices The macro-model of VAD developed in the VADFEM code is dedicated to prediction a stress and strain states in used materials after an active load. Besides, the program allows simulation a partial or/and a total unloading of the VAD’s material, what plays crucial role while applying cyclic load.

The experimental validation of developed VADFEM code on a macro-scale is achieved by comparison with distributions of a strain and a displacement measured in a digital image correlation (DIC) during deformation induced by a set of static blood pressures.[16,32] The numerical simulations are carried out in conditions given in the experiment to compare the results obtained on an external surfaces of the blood chamber of VAD. The physical model of VAD used in the

Fig. 5. (a) Error of the effective Young's modulus versus number of iterations for different relaxation coefficients, (b) number of iterations versus different relaxation coefficients. Reproduced with permission.[11] Copyright 2012, Elsevier.




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In the special case, when v ¼ 1, the relaxation iterative method goes into the standard iterative method, what for the effective Young’s modulus leads to:

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Fig. 6. Effective strain in the VAD's prototype – the POLVAD_EXT: (a) the ABAQUS code and (b) the VADFEM computer program. Reproduced with permission.[15] Copyright 2011, Wroclaw University of Technology.

Table 1. The maximum values of selected parameters computed for the macromodels of the POLVAD and the POLVAD_EXT in the VADFEM and the ABAQUS computer programs.

POLVAD

POLVAD_EXT

Parameter

VADFEM

ABAQUS

VADFEM

ABAQUS

Effective strain Effective stress [kPa] Pressure [kPa]

0.00016 58

0.00019 68

0.00011 33

0.000105 35

31

35

14

16

experiment is shown in Figure 1b before painting and in Figure 3a after painting. (Speckle pattern of surface of the blood chamber needed in DIC’s measurement is caused by mechanically hand-applied paint.) The examined polymer applied in the VAD is sensitive to changes of temperature and this observation is considered in all prepared numerical models. The comparison of experimental and numerical results shows an acceptable quantitative (Table 2) and qualitative coincidence, especially under the biggest pressure and temperature, especially for the distributions of displacement (Figure 7a and b).

Table 2. The relative displacement jUZ j, the maximum ex,MAX and minimum ex, values of strains reached in the DIC’s experiment and in the VADFEM code under the biggest pressure 37.3 kPa and in two temperatures (25 and 38 °C).

MIN

DIC Parameter

25 °C

|UZ| [mm] ex,MAX ex,MIN

1.0 0.01 0.004

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There are some heterogeneous distributions of strains in the DIC’s experiment.[33] Analysis of the results revealed that there are errors in the obtained data (Figure 7c and d). The sources of errors are distinguished as follows:

VADFEM 38 °C

25 °C

38 °C

1.3 0.013 0.007

0.72 0.006 0.0038

1.4 0.012 0.007


1) Measurement errors in the DIC’s experiment. The roughness on the surface of the blood chamber is due to hand application of the paint. The paint on the blood chamber surface is not perfectly regular; it dried and, after a certain period of time, it fell off. The surface of blood chamber is not flat; it contains curves and irregularities that can be viewed in the camera images. It is not possible to track the exact same point location on such an irregular surface. The blood chamber is permanently fixed in the experiment, but it is difficult to completely eliminate its movements during the experiment. These movements are especially visible in the small strain distributions that are measured by DIC at 10 kPa, as they have larger strain distribution irregularities at this value. 2) The blood chamber material is temperature sensitive. The blood chamber could not be heated to a uniform temperature (37 °C) using a hydraulic water system. Furthermore, a homogenous temperature distribution cannot be obtained on the blood chamber's external surface, because the device is still in contact with environment (20–25 °C are typical environmental temperatures for working LVADs). The blood chamber cannot be perfectly deformed in all directions by applying pressure on its internal surface. Consequently, a uniform strain distribution cannot be obtained on the external surface. 3) The FE model of the blood chamber required the solution of a nonlinear elastic problem. 4) This solution introduces a certain computational error that is typical for nonlinear tasks. 5) The standard deviation of the displacements is less than 2%, as computed by the DIC software. The standard deviation of the strains is less than 20%. The typical and representative error distributions (standard deviation) are




REVIEW Fig. 7. The distribution of displacement in Z direction on an external surface of the blood chamber of the VAD under pressure 37.3 kPa and in temperature 37 C: (a) the DIC's result, mm, (b) the FE model's result in the VADFEM code, mm, (c) standard deviation of the DIC's Zdirectional displacement (in mm), and (d) standard deviation of the DIC's Xdirectional principal strains (shown values × 10 − 3). Reproduced with permission.[33] Copyright 2014, Techno Press.

shown in Figure 7c and d for the errors in the Z-directional displacements and the X-directional principal strains at a pressure of 37.3 kPa. 2.7. Identification of Mechanical Properties of Polymers The polymers (thermoplastic polycarbonate urethanes) applied in the Polish VADs are Chronothane 55D – POLVAD, ChronoFlex C 55D – POLVAD_EXT and Bionate II 55D – Religa Heart Ext.[10,13,16] The polymers are examined in tension tests in two temperatures (21 and 38 °C) at the Foundation of Cardiac Surgery Development in Zabrze in Poland. The specimens are prepared in an injection method and tested on the machine MTS Criterion Single equipped with a force sensor 5 kN. The registered data is developed in the MTS TestWorksTM software. The tests and calculations are performed according to the standards: ISO 527–2 and ASTM D 638. The effective Young’s modulus is introduced in a solution of the FE problem in the first macro-models of VADs’ ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

prototypes: the POLVAD and the POLVAD_EXT. Because of this fact, values of elastic properties limited by the Young’s modulus are high and a stiffness of designed material is also quite high. This allows to expect small values of computed stresses and strains in the model examined under a real load (the maximum pressure of blood in a left chamber). Assuming, a linear elasticity of materials is a simplification, because the real material model of polyurethane is a nonlinear elastic. For older prototypes of the Polish VADs (the POLVAD and the POLVAD_EXT) without the TiN coating, it is impossible to reach the stress values higher than these of a linear elastic range measured for polycarbonate urethanes. Additionally, the stresses are not a function of velocity in ranges of values reached in the walls of blood chamber. Therefore, a simplification of material models is really advantageous for older prototypes of the Polish VADs tested in the first versions of VADFEM code, since it limits a computing time and does not introduce an error caused by a wrong assumption of the material model.



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M. Kopernik/Development and Application of Multi-Scale Numerical Tool … The following effective Young’s moduli and Poisson’s moduli are assumed for the VAD’s polymers: 1) Chronothane 55D: E ¼ 423 MPa and n ¼ 0.4;[10] 2) ChronoFlex C 55D: E ¼ 470 MPa and n ¼ 0.4;[13] 3) Bionate II 55D: E ¼ 200 MPa and n ¼ 0.4.[16] In the latest author’s works performed for materials of the Religa Heart Ext,[13,23] the identified Hollomon’s material model of Bionate II 55D is applied as 36.45e0.72. 2.8. Optimization of Model of Prototype of Ventricular Assist Device The multi-objective optimization of biomedical models using a commercial software is a well-known approach.[34] The numerical and experimental validation of results computed for the Polish VAD affirmed a correctness of the VADFEM computer program. Considering robustness of algorithms implemented in the VADFEM software and a short computation time needed for a 3D nonlinear solution, the multi-objective optimization of shape parameters of the Polish VAD is performed. The selected shape parameters are: a thickness, h, and a distance, d, between connectors of the VAD, under the maximum blood pressure (37.3 kPa) and a temperature (37 °C). The minimum of goal function, f, is determined in FEs of the macro-models of VAD as follows: f ¼ ei k ¼ ei

s0 ! min si

ð8Þ

where k is a triaxiality factor, ei is an effective strain, s0 is a mean stress, and si is an effective stress. The selection of goal function is associated with a character of damage, which can be observed in the multi-layer model of VAD loaded by a blood pressure. The relation between stress triaxiality and stress damage due to a set of loadings is widely described in literature.[35] The greatest values of triaxiality factors are computed in FEs on the inner surface of the blood chamber between connectors. The sets of CAD and FE models of the Polish VADs are prepared for the purpose of optimization. The relationship between the goal function, an effective strain, and a triaxiality factor is plotted in

Figure 8a. The relationship between the goal function and the shape parameters is shown in Figure 8b. The analysis of results obtained by the macro-models shows that a distance between the connectors is the most influential control parameter of the goal function. In conclusion, the best design of the Polish VAD with respect to the shape parameters is represented by the model 4, which has the thickest walls and the smallest distance between the connectors. The worst design of the Polish VAD, with respect to the shape parameters, is represented by the model 5, which has the thickest walls and the largest distance between the two connectors. One of older prototypes of the Polish VADs represented by the model 3 does not differ very much from the best mechanical design with respect to the defined goal function. The character of local distributions of effective strain and triaxiality factor is presented on the example of distributions computed for the best design of one of older prototypes of the Polish VADs and it is shown in Figure 9. The maximum of effective strain (Figure 9a) is observed in the FEs between the connectors. The maximum of triaxiality factor (Figure 9b) is computed for the FEs in the region closer to the center of the connectors. The selected optimization parameters should be incorporated in development of future and better designs of prototypes of the Polish VADs. The proposed approach is to be considered before FSI (fluid structure interaction) analysis. Thus, a blood flow in prototype of the Polish VAD will be examined in the safest model with respect to the goal function, in which a probability of damage is significantly reduced.

3. FE Model on Micro-Scale 3.1. Bases of FE Model on Micro-Scale The analysis of strain distributions on inner surface of blood chamber in a FE macro-model is helpful to determine correctly areas with the biggest tendency to failure. Therefore, only these places are further taken into account and investigated in a micro-model of VAD’s wall. Subsequently, a strain state reached during loading in the macro-model of blood chamber is introduced into the micro-model of VAD’s

Fig. 8. The goal function versus: (a) an effective strain and a triaxiality factor and (b) shape parameters of the Polish VADs. Reproduced with permission.[17] Copyright 2013, Wroclaw University of Technology.

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REVIEW Fig. 9. Local distributions of (a) an effective strain and (b) a triaxiality factor in the model 4 of prototype of the Polish VAD. Reproduced with permission.[17] Copyright 2013, Wroclaw University of Technology.

wall. The analogous micro-model enriched with boundary conditions taken from experimental nanoindentation test and micro-tension test is shown in the present work. Thus, the developed FE model of wall of VAD on a micro-scale is used to:

to an item, to which belongs a nanostructure. The latter assumption means that averages of all gradients vanish in the RVE. The method is based on a principle of volume averaging, leading to a definition of a macro-stress tensor in the form:

1) Identify mechanical properties of the TiN coating deposited on polymer by using results of nanoindentation test.[12,18,19] 2) Perform sensitivity analysis of the developed micro-model of mechanical test.[12,21,22] 3) Model deformation of the wall of VAD – a part of the multi-scale model of the blood chamber.[20] 4) Identify fracture parameters of the wall of VAD in a tension test in a micro-chamber of scanning electron microscopy (SEM).[13,23]

s ¼

In all of the mentioned aspects, a big number of simulations is needed. This fact is caused by a need of sensitivity (1), inverse (2), and factor analysis (3), and a need of many computations of a stretched specimen (4). This is a justification for choice of the FEM to solve the problem among other wellknown methods on a micro-scale. For example, it is difficult to apply a molecular dynamics (MD) method to solve indicated practical problems, because of computational costs.[36] On the other hand, the FEM is often used for simulation of a microand nano-scale deformation problems,[37–39] therefore, it is justified to perform an inverse analysis, a sensitivity, and a factor analysis in a nano-scale with an acceptable accuracy. The developed micro-model of TiN/polymer in the VADFEM code enables introduction and identification of parameters as follows: a residual stress,[15,23] a failure strain,[13] shape parameters of a wave character of a coating (a thickness, an antinode, and a wavelength),[20] material models of the TiN and the polymer,[12,16] and then, an analysis of their influence on an output data.[20,22] The mentioned set of parameters is crucial for an occurrence of phenomenon of a loss of cohesion observed between the TiN nano-coating and the polymer. The developed micro-model of VAD’s wall is based on a RVE composed of a substrate (polymer) material layer and a deposited very thin TiN nano-coating. The RVE is an intermediate scale between a nano and a macroscopic ones.[40] Inhomogeneities of a nanostructure are supposed to be small compared to the RVE, and the RVE should be small compared ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

Z

1 V RVE

ð9Þ

s dV V RVE

where s is the average stress; integration is carried out over the RVE with a volume VRVE. The theory of small elastic–plastic deformations is considered in modeling of the TiN nano-coating. The well-posed problem on a micro-scale also requires an equality of macrowork with an average volume of a micro-work: s e ¼

1 V RVE

Z ð10Þ

se dV V RVE

which is known as the Hill–Mandel macro-homogeneity condition,[41] where e is the average strain. Equation 9 is satisfied by three types of boundary conditions on a microscale: static, kinematic, and periodic. In the periodic boundary conditions, a displacement components perpendicular to a side of the RVE are imposed, but ones parallel to the side are let free. In this case, each side of the RVE is a symmetric plane. According to literature,[42] it appears that the best are periodic boundary conditions using an average computed on the whole mesh. The nonlinearity of mechanical properties of the TiN nanocoating and the polymer are observed. Thus, an elastic–plastic and a nonlinear elastic material models, as well as their corresponding theories, are used in computations. The microscale boundary problem included an unloading process is solved by the micro-model composed of 3700 four-node FEs and 3800 nodes. The boundary conditions, displacements, and meshes applied in the FE micro-models of mechanical tests are shown in Figure 10. 3.2. Identification of Mechanical Properties of TiN Nano-Coating The model of a multi-step nanoindentation test, accounting for a residual stress distribution, is introduced in the present



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Fig. 10. (a) The boundary conditions, a displacement and a mesh applied in the FE model of nanoindentation test. Reproduced with permission.[22] Copyright 2014, Elsevier. (b) The boundary conditions, a distribution of normal strain exx and a mesh used in the FE model of microtension test. Reproduced with permission.[23] Copyright 2014, Hindawi.

section. The specimen is composed of a ferritic steel and a biocompatible TiN coating deposited by a pulsed laser deposition (PLD) method. However, a wall of VAD is composed of a TiN nano-coating deposited on a polymer. Because of difficulties of precise measurement of force– displacement data for the soft substrate of polymer and for the very thin hard nano-coating of TiN in a nanoindentation test, it proved in literature that correctness of results measured for these coatings deposited on a steel is better.[22] The sensitivity of results of nanoindentation test reached for different substrates with respect to properties of TiN influences on accuracy of determination of mechanical properties in an inverse analysis also applied in the present section. The use of a steel instead of a biopolymer as a substrate for measurement of properties of the TiN increases an accuracy of determination of plastic properties of the TiN nano-coating in the nanoindentation test without significant reduction of an accuracy of determination of elastic properties. The justification of assumptions applied in a micro-model of nanoindentation test introduced in the present section is based on results of a sensitivity analysis conducted in literature.[12,18,21,43,44] The sensitivity analysis of a model of nanoindentation test with respect to shape parameters of an indenter (a radius and an angle of the Berkovich indenter), [21] parameters of the Coulomb friction law and elastic properties of a deformed single thin layer of material is done in a commercial FE code in literature.[43] The results show that the nanoindentation test is the most sensitive to the shape parameters of the indenter. The models of VAD’s materials (nano-coatings and polymers) used in constitutive laws are bilinear elastic–plastic models,[21,22] the Hollomon’s material model and an exponential material model.[13,16,23] The sensitivity analysis of a model of nanoindentation test with respect to parameters of an elastic–plastic material model of coating for various systems of coatings with different combination and number of coatings is investigated in a commercial FE code in literature.[18] The sensitivity analysis shows that loads (forces) are less sensitive for smaller values of a hardening exponent and bigger values of a strength index. The load is the most sensitive to the Young’s 288


modulus and slightly less sensitive to a strain-hardening exponent. The sensitivity analysis of the model of nanoindentation test in a loading and an unloading phases with respect to thickness of a deposited single coating, a residual stress and parameters of elastic–plastic bilinear material model is computed in FE codes in literature.[12] The results of sensitivity analysis unequivocally prove that small disturbances of parameters did not have big influence on a model output. The sensitivity with respect to a residual stress is larger during an unloading stage than during a loading stage. Owing to small values of sensitivities, a need for a precise evaluation of a residual stress and a thickness of coating measured in the experiment is not so restrictive, but they evaluation cannot be neglected. The micro-model of nanoindentation test is based on experimental studies done in literature.[12] Thus, introduced parameters are: a thickness of coating (350 nm), a residual stress (1.5 GPa), an angle of the Berkovich indenter (70°), and a displacement of the indenter (200 nm). The simplifications introduced in the model, which considerably decrease computational cost and do not cause loss of important information, are an approximation of a shape of the Berkovich indenter by a cone and using a 2D axisymmetric solution instead of a 3D. The application of the 2D solution instead of the 3D is proved by simulations done in the 2D and 3D.[18] The FE simulations give similar results, therefore, the 2D axisymmetric solution is applied. The indenter is made of a diamond, but in the model it is a rigid object. The selected value of displacement of the indenter is comparable to the value of thickness of coating. Therefore, the value of displacement is chosen so as to be able to observe an influence of a substrate. Slip contact condition is applied between the indenter and the specimen. The same type of mesh is generated on both materials (fournode quadrilateral FEs). There are two fine mesh areas in the FE model of the specimen. They are located in the upper coating and in the central part of the specimen. The boundary conditions, a displacement, and a mesh applied in the FE model are shown in Figure 10a. The initial stress fs 0res g is taken into account in formulation of a boundary




3

2

1n n n 0 6 6 n 1n n 0 s i ðei Þ 6 fs g ¼ ei ð1 þ nÞð1 2nÞ 6 n 1n 0 4n 0

0

0

7 7 7feg fs 0res g 7 5

ð1 2n=2

ð11Þ where:

fs 0res g ¼

8 s 0res > > > > > < s 0res

9 > > > > > =

> s 0res > > > > : 0

> > > > > ;

ð12Þ

where fs g ¼ f s r s z s u trz gT is the stress tensor in a cylindrical coordinate system, feg ¼ f er ez eu 2erz gT is the strain tensor in a cylindrical coordinate system, n is the Poisson’s ratio, and {s0res} is the initial (residual) stress. The relationship si(ei)/ei in Equation 11 is a plastic modulus, which is equal to the Young’s modulus in an elastic zone for a linear elasticity. The method of an iterative calculation of the modulus of plasticity is used for an iterative linearization of material models.[11] The variational principle of a nonlinear elastic and an elastic–plastic theory leads to a functional form for a FE e: Z

We ¼

1 fUgT ½BT ½D½BfUgdV 2 Ve Z Z T fUgT ½BT fs 0res gdV fUgT N p dS Ve

ð13Þ

Se

where S is the contact surface between the specimen and the indenter, {p} is the contact stress, {U} is the displacement vector, {U} is the matrix containing derivatives of shape functions, [D] is the matrix containing appropriate material is the matrix of shape functions of a FE, V is the properties, N volume, Ve is the volume of a current FE e, Se is the contact surface between current outside element e and a surface of the indenter. A stiffness matrix [K] and a load vector {F} are written according to the forms: Z ½BT ½D½BdV

½K ¼

ð14Þ

Ve

Z Z T T ½ B s p dS F ¼ N f 0res gdV f g V

ð15Þ

Se

The initial stress (residual stress) distribution has to be specified for each element of the mesh in the beginning stage ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

of a solution process. The value of initial stress in the TiN coating is interpreted as a mean stress and is computed by applying an inverse method,[12,15] a XRD analysis,[12,45] and profilometric studies.[23] Modeling of an unloading is realized according to a theorem of unloading. The algorithm is performed for each time step of the unloading: 1) The distributions of stresses fs g and strains feg before unloading are solutions of the loading problem. 2) Then, the specimen is loaded by an inverse force and a material is modeled as an elastic. 3) The fields of fs e g and fee g are obtained after this procedure. 4) The actual state of a stress and a strain during the unloading stage is reached as a sum of the solutions:

s unload ¼ fs g þ fs e g

ð16Þ

eunload ¼ feg þ fee g

ð17Þ

where fee g is the elastic part of a strain, fs e g is the elastic part of a stress. The contact condition between the specimen and indenter is verified as follows: s unload 0 n

ð18Þ

If the condition (Equation 18) is not satisfied for a node, the contact in this point (node) is broken. If the described condition is not satisfied in all nodes, the process of unloading is finished. In conclusion, the developed model well reproduces experimental conditions, behavior of the investigated specimen, and allows to get result in a reasonably short time. It enables to reach a purpose of this part of the work, which is identification of the material model parameters for the coating deposited on substrate on the basis of the nanoindentation test data applying an inverse method.[46] The idea of an inverse analysis presented in Figure 11a for a similar case of study is to find,[19] using an optimization procedure, the parameters of material model, which give the best matching between results of a FE simulation and an experiment. Thus, the specified goal function is a mean square root error between measured and predicted loads in a nanoindentation test: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X w¼t ðFi FiSIM Þ2 N i¼1 EXP

ð19Þ

where FEXP is the experimental forces versus depths, FSIM is the values of forces versus depths predicted by a FE model, and N is the number of sampling points in each test. The inverse procedure based on a simplex optimization technique allows to determine parameters of the material model of the TiN coating deposited on a steel in literature.[47] The following values of parameters of bilinear elastic–plastic



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problem. The relation between stresses and strains is written using a matrix form:

REVIEW


Fig. 11. (a) The flow chart of an inverse analysis. Reproduced with permission.[18] Copyright 2009, Elsevier. (b) The force–displacement data of the nanoindentation test and the model of test, in the left upper part – a bilinear material model of the TiN. Reproduced with permission.[15] Copyright 2011, Wroclaw University of Technology.

material model of the coating are obtained (Figure 11b): e1 ¼ 0.009, s1 ¼ 2614 MPa, e2 ¼ 0.166, and s2 ¼ 9107 MPa, which lead to the Young’s modulus equal to 290 GPa. The obtained elastic modulus is in a range of values given in a review shown in literature.[12] The difference between the resulting curves shown in Figure 11b can be caused by a low sensitivity of force with respect to the mechanical properties of the TiN. It can also be an error in estimation of the properties of the substrate. Therefore, author of the paper will repeat the nano-test for bigger number of samples composed of the TiN deposited on polymer. The main advantage of the selected bilinear material model of TiN is that it does not introduce the lack of uniqueness in solution, because only one set of parameters gives the minimum of the goal function (equation 19). 3.3. Analysis of Strain and Stress States in Multi-layer Wall of Prototypes of Ventricular Assist Devices The micro-model of a wall of VAD is a part of the multiscale model of VAD and is similar to the micro-model of a

nanoindentation test, but additionally the micro-model of the VAD’s wall is coupled with the macro-model of VAD and contains a roughness of surface of nano-coating described by a sinusoidal function.[11] The roughness of coating is not applied in the micro-model of a nanoindentation test, because the dimensions of roughness of surface in relation to the dimensions of the indenter are very small. In the present section, the roughness of nano-coating is approximated by a surface wave defined using an antinode and a wavelength of nano-coatings represented by a TiN deposited on a polymer by physical vapor deposition (PVD) process.[48] The micromodel of Polish VAD is developed using experimental data presented in Figure 12a. The values of an input parameters (a thickness h, a residual stress s0, parameters of a surface wave: an antinode A and a wavelength L) of the micro-model are differentiated according to a factor analysis.[49,50] The RVE is composed of the polymer and the TiN nanocoating. Considering such phenomena as an nonlinear elastic, as well as an elastic–plastic deformation and an unloading process, a boundary problem on a micro-scale is solved by the FE micro-model. An initial stress {s0res} in the TiN

Fig. 12. (a) The TEM's microstructure of 350 nm of the TiN coating deposited on the polymer by the PLD method; The distributions of (b) a mean stress and (c) an effective strain for the micromodel of the TiN coating (L = 150 nm, A = 50 nm, h = 350 nm, s 0 = 1 GPa) and the polymer of the POLVAD_EXT. Reproduced with permission.[20] Copyright 2011, Wroclaw University of Technology.

290





fs m g ¼ ½Dm fem g s m 0res

ð20Þ

m and em are the where s m 0res is the residual stresses; s stress and a strain tensors in a vector format. The variational principle, the effective Young’s modulus, the stiffness matrix, and the load vector are formulated as it is introduced in the Equation 1 and 13–15. The following types of boundary conditions on a surface of the RVE can be imposed in the analyzed case of study: Kinematic boundary conditions: M Um i ¼ eij xi

ð21Þ

Static boundary conditions: M ni s m ij ¼ ni s ij ;

ð22Þ

M where Um i is the distribution of displacements; s ij is the macroM stress and eij is the macro-strain tensors corresponding to a certain point XM in the macro-model, and evaluated directly by an average volume of a micro-stress s m ij and a micro-strain M em ij . An average stress inside the RVE is equal to s ij . Periodic boundary conditions: a displacement component perpendicular to the side of the RVE is imposed, but ones parallel to the side are let free. In this case, each side of the RVE is a symmetry plane. A problem solved by applying different types of boundary conditions will not predict the same behavior of the RVE.[28] In the considered case, all types of boundary conditions do not provide all required information on a micro-scale analysis. Generally, Periodic Boundary Conditions (PBC) are usually preferred, because they provide the most reasonable estimation of mechanical properties in heterogeneous materials, even if the structure of RVE is not periodic.[42] In the present section, the PBC are used in the parallel direction of a surface of the blood chamber and the static BC are applied in the direction of a blood pressure. Considering the TiN nanocoatings of Polish VADs, the following modification of PBC is proposed. The deformation tensor is taken from a macro-scale in modeling the boundary conditions of the RVE. The direction of normal to a surface of RVE for the TiN nanocoating corresponds to the direction of a hydrostatic pressure p (a blood pressure pm). Therefore, shear stresses and strains are not present on this surface and a stress p is one of principal stresses of a stress tensor. Thus, the solution in main coordinate system is found. The principal strains can be used as boundary conditions. Each side of the RVE is a symmetry plane with these boundary conditions, because the shear stresses and strains are zero. The 21/2 boundary problem is considered. The strain eM 2 (second principal strain of a strain tensor) is introduced into a micro-scale as a constant. Therefore, the 3D boundary problem of the RVE deformation


is transformed to the 2D plane strain problem with a current M m M value of eM 2 . The principal strain e1 and pressure p ¼ p ¼ p are also used in the model of RVE. Concluding, the boundary conditions are moved from a micro-scale to a macro-scale at this stage of modeling. The numerical experiment is planned based on a factor analysis.[50] The 132 simulations are performed for the micromodels built for two prototypes of the Polish VADs (POLVAD and POLVAD_EXT) in a loading (16 kPa) and an off-loading conditions. The micro-model is used exactly in these regions where in the macro-model of VADs the maximum values of strain and stress are observed – between two connectors (the most probable failure-source regions).[10,15] The following maximum values of strains computed in the failure-source regions are introduced into the micro-model: 1) POLVAD: e1 ¼ 0.0002; e2 ¼ 0.0002; p ¼ 16 kPa; 2) POLVAD_EXT: e1 ¼ 0.0001; e2 ¼ 0.00005; p ¼ 16 kPa. The maximum values of computed effective strain and mean stress considered as crucial parameters of a phenomenon of loss of cohesion are scanned from critical points of the micro-model (wave nodes of a surface of the nano-coating). The analysis of values of input parameters (h, s0, A, and L) in conditions without a blood load shows that the most influential parameters for the micro-model of TiN/polymer are a thickness h and a residual stress s0. Contrary, a period L is less influential parameter with respect to strains reached in a simulation. The period of wave of a surface of the TiN is more influential with respect to computed stresses. The analysis of a local distributions of crucial parameters of a phenomenon of loss of cohesion for the micro-model of TiN/polymer shows smaller gradients for thickercoatings.The densersurfaceofcoatings(smallerperiods) introduces smaller values of computed crucial parameters (“spring effect”). Therefore, a probability of occurrence of a phenomenon of loss of cohesion will be observed more often for thicker coatings, because their values of compressive mean stresses are similar to the values of tension stresses. There is no difference between a character of local distribution of selected parameters for the POLVAD and the POLVAD_EXT, differences are only observed in values of the computed crucial parameters. The smaller values of analyzed parameters are reached in the micro-model of POLVAD_EXT. The biggest gradients of stresses are observed in the nano-coating andon boundaries between the TiN and the polymer in the micro-model what can lead to fracture. The selected distributions of a mean stress and an effective strain for the micro-model of TiN/polymer of the POLVAD_EXT are shown in Figure 12b and c. The selected results of crucial parameters taken from critical points of the micro-model of the POLVAD_EXT are plotted on Figure 13. The developed micro-model of the wall of VADs in the VADFEM computer program is helpful to reach the distinguished goals: Simulation of stress and strain states on a micro-scale for failure-source regions observed in the macro-model of the blood chamber of VADs.



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nano-coating is taken into account in the FE formulation. A relation between stresses and strains is written using a matrix (vector) definition:

REVIEW


Fig. 13. The selected results of crucial parameters reached in a factor analysis computed for the micro-model of the TiN (h ¼ from 50 to 350 nm; A ¼ 50 nm, L ¼ 150 nm, s0 ¼ from 0.5 to 2 GPa) deposited on the polymer of the POLVAD_EXT.

Analysis of the crucial parameters of a phenomenon of loss of cohesion. 3.4. Identification of Fracture Parameters of TiN NanoCoating The Polish VAD is made of a thermoplastic polycarbonateurethane and a TiN nano-coating. A biocompatible thin film is recommended for surfaces of the VAD, so the design has the TiN nano-coating deposited by a PLD technique in a room temperature. Some disadvantages of this structure are presented in literature.[49] The PLD is a low temperature process allowing permanent modification of some properties only in superficial layers of product, so it can be successfully applied on the polymer substrate. However, this technique similarly as other PVD processes usually develops high values of residual stresses.[51] The residual stresses in a thin film system can lead to: buckling, cracking, a void formation, and a film debonding.[52] For a room temperature deposition, stresses easily can reach an unacceptable level.[49,53] Fortunately, they can be reduced by a selection of process parameters,[53,54] and/or by a selection of composition of deposited layers.[55–57] So far a two-scale model of the medical device has been developed,[10,11,14] it is enriched with an experimental data related to material properties,[10,12,13,16] the design related phenomena observed both on a micro-scale,[20,22] like cracks, fracture, etc. and on a macro-scale, e.g., efficiency of the model shape.[15,17] The multi-scale analysis in the VADFEM code confirmed a possibility of fracture,[15,33] because of varied mechanical properties of the thin film and the substrate,[10,11,14] as well as presence of a residual stress in the coating.[12,45] Therefore, an identification of fracture parameters: a failure strain and a residual stress is the purpose of the present section.[13,23] 3.4.1. Failure Strain Progress has been achieved in evaluating a toughness of hard coatings and thin films over the past decade.[58] The developed methodologies are based on indentation, bending, and micro-tensile testing. The typical literature reports 292


present an approach to experimental micro-tension test, in which a strain and stress curves and SEM’s microphotographs of specimens are obtained.[59–61] The second group of literature studies is dedicated to numerical modeling of in situ SEM’s micro-tension/micro-compression test, in which a FEM and a discrete dislocation dynamics (DD) are applied.[62,63] According to the latest trend, the micro-tensile testing and the FE model of the test are applied in present section to determinate fracture parameters for the TiN coatings deposited on the Bionate II. The TiN nano-coatings of thicknesses 50 and 100 nm are deposited on the Bionate II substrates by using pulsed Nd: YAG laser system proposed at the WIMiIP AGH.[54] The developed parameters of deposition process are: 100 mJ energy of laser beam, 266 nm wavelength, 4.2 J cm2 fluence, 25 °C temperature of substrate, 12 ns pulse duration at a repetition rate of 10 Hz, 10 000 laser impulses for the TiN coating of thickness 100 nm and 5000 laser impulses for the TiN coating of thickness 50 nm. The micro-tension test for the both compositions of specimens is performed directly during the SEM’s observation. During tensile tests, the samples are elongated by steps, and tensile force in a range from 0 to 40 N and elongation from 0 to 132 mm are applied. After each step of elongation, the surface of deformed thin film is observed using SEM in order to detect cracks appearance. Behavior of the coating during tensile is modeled using the VADFEM code. Additionally, the thickness and roughness of the TiN coatings are measured by atomic force microscopy (AFM). The independent verification of thicknesses of deposited coatings is done in transmission electron microscopy’s (TEM) studies. The micro-scale models of the TiN/polymer developed in the present paper have irregularities (roughness) due to the TiN nano-coating and they are based on the AFM’s results. The roughness of a nano-coating is often approximated by a periodic function with three key parameters: an amplitude, a wavelength, and a thickness.[48] This approach is realized in the developed numerical models on a micro-scale.[20] Hence, the average values of antinode and wavelength are calculated based on a spectral roughness results (“the raw AFM’s results”) measured for the TiN nano-coatings




REVIEW Fig. 14. (a) The AFM’s image of topography of the TiN nano-coating of thickness 50 nm on the selected line of the area 100 nm 100 nm; (b) the AFM’s image of topography of the TiN nano-coating of thickness 100 nm on the selected line of the area 1 mm 1 mm.

(thicknesses: 50 and 100 nm). The average wavelengths of coatings are calculated based on “the raw AFM’s results” for the areas 100 nm 100 nm (TiN of thickness 50 nm) and 1 mm 1 mm (TiN of thickness 100 nm) in many places of samples for several line scans. The AFM’s image as the representative of the coatings’ topography on the selected line scan of the area 100 nm 100 nm for the TiN coating of 50 nm is shown in Figure 14a. The AFM’s image as the representative of the coatings’ topography on the selected line scan of the area 1 mm 1 mm for the TiN coating of 100 nm is shown in Figure 14b. Thus, the micro-model of sample based on the AFM’s results represents a character of a wave of the TiN coating, not a wave character of whole surface of the substrate. It is a local micro-model. The average calculated parameters of a sinusoidal surface of the TiN nano-coatings are: (i) an antinode – 12.5 nm and a wavelength – 100 nm for TiN coating of thickness 50 nm, (ii) an antinode – 20 nm and a wavelength – 150 nm for TiN coating of thickness 100 nm. The same micro-model as in Section 3.2 enriched with boundary conditions taken from the experimental microtension test is developed in the present section. The micromodel incorporates the experimental parameters: a residual stress, shape parameters of the coating wave’s characteristics, material models of the TiN coating, and the polymer. The relation between stresses and strains, the variational principle, the effective Young’s modulus, the stiffness matrix, and the load vector are formulated as it is introduced in the Equation 1, 13–15, and 20. The periodic boundary conditions are used and the kinematic boundary conditions are applied in a direction of tension. The deformation in the direction of tension eM 1 (the first principal strain of a strain tensor) obtained in the in situ SEM’s micro-tension test as the mean X-strain is used as a boundary condition for the RVE. Strain eM 2 (the second principal strain of a strain tensor) taken from the in situ SEM’s micro-tension test as the mean Y-strain is introduced in the micro-model. Therefore, a 3D boundary problem of the RVE deformation is transformed to a 2D plane strain problem with a prescribed value of the strain eM 2 . The methodology of extraction parameters of a facture model based on the in situ SEM’s micro-tension test is ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

presented in literature.[64] The key parameter, which represents a fracture is called a ductility function, c. Generally, this parameter is defined by the formula: c¼

ei ep

ð23Þ

where ei is an effective strain, ep is a critical deformation. The critical deformation ep is based on the experimental tension tests at the moment of a fracture occurrence and can be a function of temperature, thickness of coating, and parameters of a surface profile. Determination of the ductility function’s c parameters based on the in situ SEM’s microtension test for the two types of specimens is the goal of simulations on a micro-scale. The observations based on the in situ SEM’s micro-tension test indicate that the deposited film cracks only after an external forced deformation and the cracks are arranged at an angle of 75–85° to a stretching direction. There are some small cracks on the surface of the coating at the middle stage of deformation (a tensile force of 15 N, an elongation of 2 mm). When the sample is stretched at about 0.2 mm, the number of cracks begins to increase rapidly. When the sample is stretched at exactly 0.2 mm, the entire sample surface is covered with cracks. It should be noted that extension of the TiN layer of thickness 50 nm of about 0.2 mm causes very big deformation. The numerous cracks occur at the stretching of several mm. The cracks propagate in the perpendicular direction to the main cracks and are caused by a shrink and a deformation of the substrate in the direction perpendicular to the stretching. The justification of cracking of the TiN coatings similar to the observations made in the present section is considered as a mechanism of wrinkling and it is shown in literature.[53] Based on settings of the experimental micro-tension tests, the corresponding numerical models of the test are prepared on a micro-scale for the TiN coatings (thicknesses: 50 and 100 nm). The simulations are performed for the initial, the middle and the final stages of the test, which corresponds to the forces 8 N (exTiN ¼ 0.0011903), 15 N (exTiN ¼ 0.0028368),



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M. Kopernik/Development and Application of Multi-Scale Numerical Tool … and 40 N (exTiN ¼ 0.0109978). These stages are selected, because the noticeable changes on the surface of samples are observed in relation to the previous steps of deformation. The numerical results represented by an effective strain and a mean stress are shown in Table 3. In the Table 3, the mean value of X-strain is obtained in the experimental tests. The SEM’s images and numerical FE’s results computed at the middle stages of the tests are presented in Figures 15 and 16. The cracking occurrence observed in experiments corresponds to the FE’ effective strain equal to 0.008514 for the TiN coating of thickness 50 nm and to the FE’s effective strain equal to 0.008117 for the TiN coating of thickness 100 nm. In conclusion, the set of fracture parameters (c a ductility function’s parameters): ei and ep for the TiN coatings of thicknesses 50 and 100 nm are identified.

3.4.2. Residual Stress 3.4.2.1. Identification of Residual Stress by Curvature Measurement Method The residual compressive stress is one of the fracture parameters identified in the TiN coatings deposited on the polymers of the Polish VADs by the PLD method.[23] The methods of residual stress measurement, including the X-ray methods, are useful, but they have some disadvantages and constraints specified in literature.[65] Measurements of a diffraction peak displacement and a broadening can be used for determination of stresses present in a near surface layers of a crystalline materials.[66,67] However, these methods are dedicated to the well-crystalline materials providing a sufficiently strong diffraction peaks. It is experimentally

Table 3. The micro-tension test: the SEM’s and the FE’s results.

TiN, 100 nm Force [N] 8 15 40

TiN, 50 nm SEM, mean, X-strain, ep

FE, effective strain, ei

FE, mean stress, MPa

Force [N]

SEM, mean, X-strain, ep

FE, effective strain, ei

FE, mean stress [MPa]

0.0011903 0.0028368 0.0109978

0.005202 0.008117 0.023063

1117.9 1672.1 2902.4

8 15 40

0.0011903 0.0028368 0.0109978

0.005474 0.008514 0.024203

1011.3 1509.4 2841.7

Fig. 15. The TiN coating of thickness of 100 nm stretched at 15 N: (a) the SEM’s image, (b) an effective strain in the FE’s micro-model, and (c) a mean stress in the FE’s micro-model.

Fig. 16. The TiN coating of thickness of 50 nm stretched at 15 N: (a) the SEM's image, Reproduced with permission.[23] Copyright 2014, Hindawi. (b) an effective strain in the FE's micromodel, and (c) a mean stress in the FE's micromodel.

294





2 2 2 2 EPu hPu Pu þ EAu ðhPu þ hAu Þ hPu þ ETiN ðhPu þ hAu þ hTiN Þ ðhPu þ hAu Þ

e¼ 2 EPu hPuPu þ EAu hAuAu þ ETiN hTiNTiN

exTiN

ð26Þ

1 0 3 3 3 3 3 3 8f @ETiN ðh eÞ ðhPu þ hAu eÞ þ EPu ðhPu eÞ þ e þ EAu ðhPu þ hAu eÞ ðhPu eÞ A

¼ 2 ETiN hTiN hPu þ hAu e þ 12 hTiN 3L

sample with parameters of the analytical model of residual stress is shown in Figure 17a. The TiN nano-coatings of thicknesses ca. 50 nm are deposited on polymer by PLD process as it shown in Section 3.4.1.[54] For one lot of samples, the gold nano-coatings of thicknesses of 5 nm are deposited as ADVANCED ENGINEERING MATERIALS 2015, 17, No. 3

e0TiN ¼

2exTiN nTiN exTiN 2 nTiN ¼ exTiN 3 3



ð27Þ

ð28Þ

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interlayers between the TiN and Bionate II. The gold is deposited by a magnetron sputtering method with a discharge current 10 mA and a deposition time 5 min. The interlayers are not imposed for specimens from a second lot of samples. Deflections are measured using an optical profilometer Wyko NT9300 (by Veeco Company). The deflections of the one layer specimens are in a Fig. 17. (a) Sketch of the sample TiN/Bionate II and parameters used in the analytical model of a residual range of experimental error. The specimen stress. (b) Camera picture of the specimen TiN/Au/Bionate II taken at the side of the specimen. Reproduced with gold buffer layer indicates significant with permission.[23] Copyright 2014, Hindawi. deflection (an increase in the specimen highest deflection equal to 0.976 mm, Figure 17b) and the proved in literature that such conditions are not fulfilled in the bending direction indicates presence of a compressive case of thin layers analyzed in the present work – and the residual stress. This value is used during calculation of the stress cannot be measured using the X-ray diffraction.[23] The most interesting solutions to avoid/reduce detrimental residuals stress level. The interpretation of profilometric experiment is based on a effects of residual stress are introduction of an intermediate methodology of determination of residual stress in layers gold layer between a TiN and polymer,[55–57] because gold is a considering a deflection of samples.[74] It is assumed that at biocompatible material. It is stated that a very thin gold layer increases strength of a structure and improves of a metalthe moment of deflection of a sample, a material is in an elastic polymer adhesion.[55,56] The gold buffer layer changes a state. The formulas (24) and (25) are written for composition of materials: the TiN and Bionate II, and they are taken from the residual stress and it is difficult to anticipate an influence of Timoshenko's work.[74] The formulas (26) and (27) are written the gold buffer layer on a residual stress in a structure.[57] for composition of materials: the TiN, Au, and Bionate II based Therefore, it is assumed in the present section that the buffer on the same assumptions as it is in formulas (24) and (25). In biocompatible thin film of Au in the TiN/Bionate II material case, when the thickness of Au thin film is small (about 0.1 system will allow to determine the residual stress in a times smaller than the thickness of TiN), the interpretation of curvature measurement method (CMM), and helps to results can be done by using formulas (24) and (25). The improve a toughness of connection between the TiN and formulas (28) and (29) are helpful to obtain a volumetric strain Bionate II. The CMM is used for a stress determination in a TiN and a mean stress in the TiN, which are necessary in the FE formulation of the problem. coating deposited on a steel,[68] on a WC-Co substrate, and for an Au coating on Si substrate.[69] A two-layer bimaterial beam or plate specimen is usually used for the CMM and a ETiN h2 h2Pu þ EPu h2Pu e¼ ð24Þ curvature is induced due to a change in initial stresses when a 2ðEPu hPu þ ETiN hTiN Þ specimen is released from constraints imposed during a stress source processing. Among all of the developed models of a 1 0 relation the curvature to residual stress in the coating,[70–73] a ETiN ðh eÞ3 ðhPu eÞ3 þ EPu ðhPu eÞ3 þ e3 8f model developed by Timoshenko is simple and has noA

exTiN ¼ 2 @ ETiN hTiN hPu e þ 12 hTiN 3L restriction on layers’ thickness in the analyzed structure,[74] so it is used in the present section. The schematic sketch of a ð25Þ

REVIEW

M. Kopernik/Development and Application of Multi-Scale Numerical Tool … Table 4. The maximum and minimum values of an effective strain, a normal stress, and a shear stress computed in the FE models of micro-tension test for the samples: TiN/Au/Bionate II and TiN/Bionate II.

Force [N]

Parameter

8

sxx [MPa]

TiN/Au/Bionate II Tensile 225 Tensile 50.063 Min 0.000844 Tensile 594.18 Tensile 87.424 Min 0.00131 Tensile 3118 Tensile 398.98 Min 0.002507

sxy [MPa] ei sxx [MPa]

15

sxy [MPa] ei sxx [MPa]

40

sxy [MPa] ei

s 0TiN

ETiN ¼ e0TiN 1 2nTiN

Compressive 238.99 Compressive 50.263 Max 0.001321 Compressive 151.72 Compressive 86.938 Max 0.00432 Compressive 515.18 Compressive 404 Max 0.021241

ð29Þ

where e is the center of gravity with respect to properties; h, hPU, hAu, and hTiN are the thickness of the whole sample, Bionate II, gold, and TiN; f is the deflection of sample in profilometric studies, L is the length of sample, exTiN is the strain component in the X direction in the TiN nano-coating, s0TiN is the mean stress in the TiN, e0TiN is the mean strain in the TiN; EPU, EAu, and ETiN is the Young’s moduli of Bionate II, gold, and TiN; nTiN is the Poisson’s ratio of TiN. According to the set of parameters introduced in Equation 24–29 and their values (Sections 2.7 and 3.2): EPU ¼ 200 MPa,[13,23] nPu ¼ 0.45, ETiN ¼ 290.4 GPa,[12] nTiN ¼ 0.25, EAu ¼ 1.3 GPa,[57] nAu ¼ 0.3, f ¼ 0.976 mm, L ¼ 23 mm; the residual stress and volumetric strain calculated in the analytical model (26)–(29) in the TiN are s0 ¼ 690 MPa and e0 ¼ 0.0012 for the material system TiN/Au/Bionate II. The Au nanocoating introduced in the designed material system results in a bigger compressive residual stress and bending of the sample, thus determination of a residual stress is possible by the curvature method. 3.4.2.2. Comparison of FE Models of Micro-Tension Test for Samples TiN/Au/PU and TiN/PU Based on settings of the experimental micro-tension tests (see Section 3.4.1), the corresponding FE models of micro-tension test of the two types of samples TiN/Bionate II and TiN/Au/Bionate are developed. The applied compositions of material layers are: (i) 50 nm of the TiN, 5 nm of the Au 296


TiN/Bionate II Tensile 627.79 Tensile 73.606 Min 0.000383 Tensile 1496.1 Tensile 175.46 Min 0.00091 Tensile 3551.6 Tensile 683.96 Min 0.00352

Compressive 158.02 Compressive 68.171 Max 0.002608 Compressive 378.8 Compressive 162.55 Max 0.006204 Compressive 1486 Compressive 696.43 Max 0.023425

and 1000 nm of the Bionate II and (ii) 55 nm of the TiN and 1000 nm of the Bionate II. The boundary conditions, materials’ layers, a normal strain exx, and a mesh used in the first version of FE model are shown in Figure 10b. The same boundary conditions and a mesh are used in the second version of model. The simulations are performed for the initial, middle, and final stages of the test. The exact values of maximum and minimum values of an effective strain, a normal stress, and a shear stress computed in the FE models of micro-tension test for the samples: TiN/Au/Bionate II and TiN/Bionate II are presented in Table 4 and they are taken from distributions of selected results plotted in Figures 18 and 19. The observations made on the basis of a character of distributions and values of stress leads to a conclusion that the Au coating improves a connection between the TiN coating and Bionate II substrate, because computed gradients of stress are smaller. The analysis of a character of distribution of an

Fig. 18. A normal stress s xx in MPa for the FE models of the samples: (a) TiN/Au/PU and (b) TiN/PU, stretched at 8 N in the microtension test. Reproduced with permission.[23] Copyright 2014, Hindawi.

Fig. 19. An effective strain ei for the FE models of the samples: (a) TiN/Au/PU and (b) TiN/PU, stretched at 8 N in the microtension test. Reproduced with permission.[23] Copyright 2014, Hindawi.




4. Conclusions 1) The multi-scale FE model of blood chamber of VAD with the deposited biocompatible TiN nano-coating is developed in the non-commercial VADFEM code. 2) The developed multi-scale FE model due to solution of problem of convergence in a nonlinear tasks (a nonlinear elasticity for the polymer of blood chamber and an elasticity-plasticity for the TiN coating) enables an optimization of crucial designed parameters of the blood chamber (parameters of shape) and a determination of fracture parameters of the nano-coating. Therefore, the comparative stress–strain analysis of the prototypes of Polish VADs is performed. 3) The FE macro-model of blood chamber is validated in numerical studies by comparison with the ABAQUS commercial FE code and in experiments by comparison with the DIC data. 4) The FE micro-model of multi-layer wall of blood chamber is calibrated and validated by using results of the in situ SEM’s micro-tensile test. 5) The developed multi-scale FE model is enriched with materials research: the tensile test of VADs’ polymers, the nanoindentation test of the TiN coupled with an inverse analysis for its interpretation, the profilometric studies, and the developed analytical model to calculate a residual stress in the TiN. 6) The final result of the work is the multi-scale numerical tool for modeling the pneumatic VADs produced in Poland

Received: April 1, 2014 Final Version: May 28, 2014 Published online: June 24, 2014

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REVIEW

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