Nonlinear Viscoelastic Constitutive Equation Research Articles

Study on Nonlinear Viscoelastic Constitutive Equations for Rock

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

Numerical Simulation on Structural Integrity of Grain with Large Increased Area Ratio for Solid Rocket Motor

To evaluate the structural integrity of a type of solid rocket motor grain with a largely increased area ratio using free-loading modified double-base grain, the Schapery integral nonlinear viscoelastic constitutive equation with damage is derived, and the numerical simulation model of solid rocket motor grain is constructed. According to the mission profile of the grain during the equipment period, the numerical simulation research on the mechanical response characteristics of the grain under three typical working conditions, namely, platform vibration, underwater explosion impact, and ignition pressurization during the launch is carried out. The simulation results show that under the three typical working conditions, the grain will not produce structural damage and defects, and the mechanical response of the grain under the platform vibration and explosion impact load is far less than that under the launch ignition pressurization. The ignition pressurization should be considered in the grain structural integrity design of this type or similar structure solid rocket motor.

Hydrodynamic interaction and coalescence of two inline bubbles rising in a viscoelastic liquid

In this paper, direct numerical simulations (DNS) are performed to investigate the inline rise of a pair of three-dimensional (3D) air bubbles in a viscoelastic liquid using the volume-of-fluid approach with an adaptive mesh refinement technique. The exponential Phan-Thien–Tanner model is used as the non-linear viscoelastic constitutive equation for the liquid. The numerical model has been validated by comparison with previously published results, including the terminal velocity jump discontinuity of an isolated bubble rising in a viscoelastic fluid, when its volume exceeds a certain critical value. Focusing on the inline rising bubble pair in such a viscoelastic medium with different configurations, we found that the wake of the small leading bubble attracts a larger trailing bubble, whereas for a supercritical bubble in front of a subcritical bubble, they tend to further separate. Before reaching a critical volume, the two subcritical bubbles remain close to each other after approaching each other, forming a stable chain. For pairs containing a supercritical trailing bubble, however, a drafting–kissing scenario occurs before the bubble–bubble coalescence. The long-range repulsion and the short-range attraction due to fluid elasticity are critical to the aforementioned bubble pair interactions. Interestingly, the terminal rise velocities of the stable bubble chain and the coalesced bubble both increase with the initial spacing. The squeezing flow near the growing bubble neck seems to delay the coalescence process. The capillary wave propagating down to the coalesced bubble tip together with the extensional flow behind the stretched bubble determines the generation of satellite microdroplets along the tail of the coalesced bubble. To the best of our knowledge, this is the first 3D DNS on a bubble pair ascending in viscoelastic fluids.

Construction of PZT-5H mechano-electric model based on strain rate dependence and its numerical simulation in overload igniter application

Although the force-electricity relationship of piezoelectric ceramics under pulse stress is found to be nonlinear, there is still lacking a reliable model describing this relationship. To bridge this research gap, the mechanical and electrical responses of PZT-5H (Pb1·0[Zr0·49Ti0·46(Nb0·25Sb0.75)0.05]1.0O3) under different strain rates were investigated through uniaxial compression tests and split-Hopkinson pressure bar (SHPB) experiments with an additional electrical output measurement system. With reference to the nonlinear viscoelastic constitutive equation and the piezoelectric equation, a PZT-5H mechano-electric model considering the strain rate effects was built based on the experimental data. The incremental form of the mechano-electric model was established based on the Piola-Kirchhoff stress tensor and Green strain tensor theory. The model was numerically simulated with a user-defined material subroutine in the explicit finite element software ABAQUS. The accuracy of the material subroutine was verified with finite element models of the uniaxial quasi-static compression and the high strain rate SHPB tests. The finite element simulation results were verified with the experimental results, and the two agreed well. The material subroutine was then adopted to predict the mechanical and electrical response of the piezoelectric ceramics inside the overload igniter during projectile launch and impact. The analysis results show that the overload igniter can be stably activated within 200 μs under the launch overload and successfully ignited within 5 μs at speed as low as 50 m/s under the impact overload.

Residual strength prediction for unidirectional CFRP using a nonlinear viscoelastic constitutive equation considering entropy damage

We propose a viscoelastic/plastic constitutive equation for viscoelastic media based on irreversible thermodynamics and the viscoelastic theory. Uniaxial tensile tests on a TriA-X polyimide resin are performed at different temperatures and strain rates to determine the material parameters for the constitutive equation. In the numerical analysis, the FORTRAN programming language is used to apply the proposed equation to the matrix resin. Transverse tensile failure analysis is performed on the unidirectional carbon fiber reinforced plastic (CFRP) in constant strain rate condition after holding constant strain. In this condition, the failure strength is lower than the static tensile strength. The same tendency is observed in the three-point bending fracture test of the unidirectional CFRP. The proposed model expresses the accumulation of resin damage while holding a constant strain and predicts the residual strength and lifetime of the unidirectional CFRP.

Peridynamic model for nonlinear viscoelastic creep and creep rupture of Polypropylene

This paper presents the peridynamic numerical method for nonlinear viscoelastic creep behaviour which consists of primary, secondary, tertiary creep stages and creep rupture. A nonlinear viscoelastic creep constitutive equation based on internal state variable (ISV) theory which covers four creep stages is examined. The viscoelastic equation is substituted into material parameter in the peridynamic equation to derive a new peridynamic method with two time parameters i.e. numerical time and real time. The parameters of the viscoelastic equation is analyzed and evaluated. In validating this peridynamic method, a comparison is made between numerical and experimental data. The peridynamic method for nonlinear viscoelastic creep behaviour (VE-PD) is approved by the good similarity between numerical and experimental creep strain curves with overall difference of 10.67%. The nonlinearity of experimental and numerical data is adequately similar as the error between experimental and numerical curves of secondary stage strain rate against load is 8.022%. The shapes of fractured numerical specimen show good resemblance with the experimental result as well.

Constitutive model for shape memory polyurethane based on phase transition and one-dimensional non-linear viscoelastic

Shape memory polymers are the most essential branches of smart materials, which can be fixed at a temporary shape and recover to permanent shape in response to environmental stimuli, such as temperature, electricity, light and magnetic field. To provide a suitable mechanism for shape memory polyurethanes (SMPU) and to predict their behavior, in this study, a nonlinear viscoelastic constitutive equation was constructed based on the viscoelasticity theory and phase transition theory. A modified three element model with Kohlrausch-Williams-Watts (KWW) stretched exponential function was used to simulate the mechanical behavior of frozen and active phases to describe the viscoelastic behavior of SMPU. A connection coefficient k, which describes the connection model of frozen and active phases, was proposed to represent the relation between the effective modulus and that of each phase.

Dependence of the subharmonic signal from contrast agent microbubbles on ambient pressure: A theoretical analysis.

This paper investigates the dependence of the subharmonic response in a signal scattered by contrast agent microbubbles on ambient pressure to provide quantitative estimations of local blood pressure. The problem is formulated by assuming a gas bubble encapsulated by a shell of finite thickness with dynamic behavior modeled by a nonlinear viscoelastic constitutive equation. For ambient overpressure compatible with the clinical range, the acoustic pressure intervals where the subharmonic signal may be detected (above the threshold for the onset and below the limit value for the first chaotic transition) are determined. The analysis shows that as the overpressure is increased, all harmonic components are displaced to higher frequencies. This displacement is significant for the subharmonic of order 1/2 and explains the increase or decrease in the subharmonic amplitude with ambient pressure described in previous works. Thus, some questions related to the monotonic dependence of the subharmonic amplitude on ambient pressure are clarified. For different acoustic pressures, quantitative conditions for determining the intervals where the subharmonic amplitude is a monotonic or non-monotonic function of the ambient pressure are provided. Finally, the influence of the ambient pressure on the subharmonic resonance frequency is analyzed.

Bend stiffener nonlinear viscoelastic time domain formulation

Bend stiffeners are conical polyurethane structures used in the offshore industry to ensure a smooth transition in the upper connection of flexible risers with the floating production unit. The polyurethane employed for bend stiffeners present a nonlinear viscoelastic response that is highly dependent on the loading rate and temperature. This may lead to different flexible riser response when compared to elastic or hyperelastic material modeling. In order to quantify this effect, tensile and relaxation tests are carried out to characterize the nonlinear time dependent mechanical behavior of the polyurethane. A constitutive equation based on the modified superposition method is then presented and a procedure for material identification proposed. A numerical scheme using the state variable approach is formulated for the nonlinear viscoelastic constitutive equation finite element implementation. The top connection system, consisting of a bend stiffener and a flexible pipe segment, is represented by a large displacement beam model accounting for geometrical and material nonlinearities. A case study is carried out to compare the system curvature response calculated for the nonlinear viscoelastic and hyperelastic (Marlow strain energy potential) bend stiffener subjected to harmonic loading conditions at different frequencies. The results show the importance of bend stiffener strain rate dependency on the riser curvature response.

Analysing the large amplitude oscillatory shear of the Cross and Cross–Maxwell models with the aid of Fourier transform rheology using the example of a solvent-borne alkyd primer

This paper contributes to the theory behind the analysis of strain-controlled large amplitude oscillatory shear (LAOStrain) using Fourier transform rheology as well as Lissajous plots and Pipkin diagrams. It applies LAOStrain to both a reduced form of the Cross model (known as the “rCross model”) and a Maxwell model whose viscous part is defined by the rCross model (reffered to as the “rCross-Maxwell model” in this paper). The analytical determination of the loss modulus and higher harmonics of the rCross model as mathema- tical functions simplifies the discussion thereof. This in turn facilitates the determination of characteristic points which are directly related to material parameters and therefore enable the identification of both the material parameters of the rCross model and their physical significance. A method for identifying material parameters can be presented which do not require minimising residuals of a non-linear optimisation scheme. It was not possible to identify an analytical solution to the non-linear viscoelastic constitutive equations of the rCross-Maxwell model, which are therefore solved numerically. The characteristic points identified for the rCross-Maxwell model are based on the analytical solutions arrived at for the rCross and Maxwell models. This means that it is also possible to both identify the material parameters of the rCross-Maxwell model and determine their physical significance. The method of material parameter identification developed is explained using the example of a solvent-borne alkyd primer and the characteristic points determined for strain amplitude dependent storage and loss moduli.

Birefringence and interface in sequential co‐injection molding of amorphous polymers: Simulation and experiment

Modelings of the interface distribution and flow‐induced residual stresses and birefringence in the sequential co‐injection molding (CIM) of a center‐gated disk were carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model the frozen‐in flow stresses in disks. The compressibility of melts is included in modeling of the packing and cooling stages and not in the filling stage. The thermally induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the relaxation modulus and strain‐optical coefficient functions of each polymer. The influence of the processing variables including melt and mold temperatures and volume of skin melt on the birefringence and interface distribution was analyzed for multilayered PS‐PC‐PS, PS‐PMMA‐PS, and PMMA–PC–PMMA molded disks obtained by CIM. The interface distribution and residual birefringence in the molded disks were measured. The measured interface distributions and the gapwise birefringence distributions in CIM disks were found to be in a fair agreement with the predicted interface distributions and the total residual birefringence obtained by the summation of the predicted frozen‐in flow and thermal birefringence. POLYM. ENG. SCI., 55:88–106, 2015. © 2014 Society of Plastics Engineers

Birefringence in injection‐compression molding of amorphous polymers: Simulation and experiment

AbstractThe influence of the processing variables on the residual birefringence was analyzed for polystyrene and polycarbonate disks obtained by injection‐compression molding under various processing conditions. The processing variables studied were melt and mold temperatures, compression stroke, and switchover time. The modeling of flow‐induced residual stresses and birefringence of amorphous polymers in injection‐compression molded center‐gated disks was carried out using a numerical scheme based on a hybrid finite element/finite difference/control volume method. A nonlinear viscoelastic constitutive equation and stress‐optical rule were used to model frozen‐in flow stresses in moldings. The filling, compression, packing, and cooling stages were considered. Thermally‐induced residual birefringence was calculated using the linear viscoelastic and photoviscoelastic constitutive equations combined with the first‐order rate equation for volume relaxation and the master curves for the Young's relaxation modulus and strain‐optical coefficient functions. The residual birefringence in injection‐compression moldings was measured. The effects of various processing conditions on the measured and simulated birefringence distribution Δn and average transverse birefringence <nrr−nθθ> were elucidated. Comparison of the birefringence in disks manufactured by the injection molding and injection‐compression molding was made. The predicted and measured birefringence is found to be in fair agreement. POLYM. ENG. SCI., 2013. © 2013 Society of Plastics Engineers

Viscoelastic effects during loading play an integral role in soft tissue mechanics

Viscoelastic relaxation during tensioning is an intrinsic protective mechanism of biological soft tissues. However, current viscoelastic characterization methodologies for these tissues either negate this important behavior or provide correction methods that are severely restricted to a specific viscoelastic formulation and/or assume an a priori (linear) strain ramp history. In order to address these shortcomings, we present a novel finite ramp time correction method for stress relaxation experiments (to incorporate relaxation manifested during loading) that is independent of a specific viscoelastic formulation and can accommodate an arbitrary strain ramp history. We demonstrate transferability of our correction method between viscoelastic formulations by applying it to quasi-linear viscoelastic (QLV) and fully nonlinear viscoelastic constitutive equations. The errors associated with currently accepted methodologies for QLV and fully nonlinear viscoelastic formulations are elucidated. Our correction method is validated by demonstrating the ability of its fitted parameters to predict an independent cyclic experiment across multiple strain amplitudes and frequencies. The results presented herein: (i) indicate that our correction method significantly reduces the errors associated with previous methodologies; and (ii) demonstrate the necessity for the use of a fully nonlinear viscoelastic formulation, which incorporates relaxation manifested during loading, to model the viscoelastic behavior of biological soft tissues.

Ice-stream response to ocean tides and the form of the basal sliding law

Abstract. The response of ice streams to ocean tides is investigated. Numerical modelling experiments are conducted using a two-dimensional flow-line model of coupled ice-stream and ice-shelf flow. The model includes all components of the equilibrium equations, and uses a non-linear viscoelastic constitutive equation for ice. Basal sliding is simulated with a Weertman type sliding law where basal sliding is proportional to some power of the basal shear stress. The response of ice-streams to tidal forcing is found to be profoundly affected by mechanical conditions at the bed. For a non-linear sliding law, a non-linear interaction between the two main semi-diurnal tidal constituents (M2 and S2) can give rise to a significant perturbation in ice-stream flow at the lunisolar synodic fortnightly (MSf) tidal period of 14.76 days. For a linear sliding law, in contrast, no such modulation in flow at the MSf frequency is found. For vertical ocean tides of the type observed on Filchner-Ronne Ice Shelf (FRIS), the amplitude of the horizontal modulation in ice-stream flow at the MSf frequency resulting from a non-linear interaction between the S2 and M2 tidal constitutes can be larger than the direct response at the S2 and the M2 frequencies. In comparison the non-linear interaction between K1 and O1 tidal components is weak. As a consequence, modelled ice-stream response to mixed oceanic tides of the type found on FRIS is stronger at the MSf period of 14.76 days than at both the semi-diurnal and diurnal frequencies, while at the same time almost absent at the similar Mf period of 13.66 days. The model results compare favourably with measurements of tidally induced flow variations on Rutford Ice Stream (RIS), West Antarctica. On RIS a strong tidal response is found at the MSf frequency with a smaller response at the semi-diurnal and diurnal frequencies, and almost no response at the Mf frequency. A non-linear viscous sliding law appears to have the potential to fully explain these observations.

Frozen-in birefringence and anisotropic shrinkage in optical moldings: II. Comparison of simulations with experiments on light-guide plates

The thermally-, flow-induced and total birefringence components and anisotropic shrinkages in LGP moldings were simulated by using a combination of a CV/FEM/FDM technique nonlinear viscoelastic and photoviscoelastic constitutive equations and orientation functions, as described in Part I of this study. The simulated results were compared with measurements on LGP moldings of a polystyrene (PS) and two optical grade polycarbonates (PCs) OQ1030 and OQ3820 having low and high molecular weights. The thermally-induced birefringence was simulated by a combination of constrained and free cooling during molding. In LGP moldings of PS, the simulated thermally-induced birefringence indicated a minor variation with location in the mold plane, a parabolic shape in the core region and an increase towards the wall. Compared to the flow-induced birefringence, the thermal birefringence provided a minor contribution to the total transverse birefringence Δn12. In LGP moldings of PCs, the simulated thermally-induced birefringence showed a significant variation with location in the mold plane, nearly constant value in the core region and high value in the wall region. In LGP moldings of both PCs, the contributions of the thermally- and flow-induced birefringence to the total transverse birefringence Δn12 were significant. The effect of processing conditions on the development of the normal birefringence in LGP moldings of PCs was ranked from most to least: the packing pressure, mold temperature, melt temperature, injection speed and packing time. However, in LGP moldings of PS the packing time effect was significant due to a longer gate freezing time. Simulated and measured normal birefringence along the flow direction was in fair agreement, but simulations were unable to describe the observed birefringence maximum arising near the gate. The averaged luminance of LGP moldings exhibited some correlation with the averaged normal birefringence. LGP moldings of PC OQ1030 indicated a pronounced maximum in the simulated transverse flow birefringence in the core but a low value near the wall. In contrast, the LGP molding of PC OQ3820 showed a high simulated birefringence near the wall and a low value of maximum in the core. The simulated and measured total transverse birefringence in LGP moldings was in fair agreement. LGP molding of both PCs showed similar tendency in shrinkage variation with processing conditions. However, the thickness shrinkage was higher in LGP moldings of PC OQ3820. The effect of processing conditions on the development of shrinkage in LGP moldings of both PCs was ranked from most to least: the packing pressure, melt temperature, mold temperature, injection speed and packing time. In LGP moldings of PS, the thickness shrinkage slightly increased with increasing melt temperature and significantly increased with reducing packing time. A good agreement between the simulated and measured anisotropic shrinkages in LGP moldings at various processing conditions was observed.

Non-linear viscoelastic models for semi-flexible polysaccharide solution rheology over a broad range of concentrations

The experimental linear viscoelastic and steady-state shear data over a broad concentration range of several aqueous semi-flexible polysaccharide solutions can be quantitatively predicted by the multimode Phan-Thien Tanner, Giesekus, and extended Pom-Pom non-linear viscoelastic models using a single set of parameters. Experimental data from an aqueous solution of the polysaccharide guar galactomannan (Mw=1.12×106 g/mol), aqueous phosphopolysaccharide “viilian” solution excreted by Lactococcus lactis subspecies cremoris SBT 0495 (Mw=2×106 g/mol), and aqueous Propionibacterium acidi-propionici polysaccharide solution (Mw≈106 g/mol) are taken from literature. A master curve can be constructed by shifting data from different concentrations over the horizontal and vertical axes. A simple model to describe the concentration dependence of the relaxation time and plateau modulus over the entire concentration range is presented. Transient shear viscosity data are quantitatively and first normal stress coefficient are qualitatively predicted for the guar galactomannan solution. Due to their proven performance in finite element simulations, these non-linear viscoelastic constitutive equations could help us to improve predictive modeling of time-dependent complex flow problems for polysaccharide solutions with variations in concentration and temperature in the spatial domain using only a single set of parameters. However, as a previous step, the performance of additional rheological experiments in simple flows, i.e., transient and steady-state extensional measurements, is recommended.

Frozen-in birefringence and anisotropic shrinkage in optical moldings: I. Theory and simulation scheme

An unconditionally stable upwinding scheme was proposed to improve the efficiency of the viscoelastic simulation in molding of optical products using a CV/FEM/FDM technique. A significant computation time saving was achieved due to an elimination of subdivision of the time step as required in the conventional numerical scheme. The approach was applied to simulate the flow-induced birefringence and anisotropic shrinkage in disk moldings using a nonlinear viscoelastic constitutive equation, orientation functions and equation of state. The two-dimensional triangular finite element meshes were used in the disk cavity and the one-dimensional tubular elements were utilized in the delivery system. Good agreement was shown between the simulated pressure traces and flow birefringence in the molding using the unconditionally stable upwinding scheme of the present study and the conventional numerical scheme of the earlier study. In addition, an algorithm for simulation of the thermal stresses and birefringence in moldings using linear viscoelastic and photoviscoelastic constitutive equations was presented by combining constrained and free quenching approaches. The proposed numerical scheme for viscoelastic simulation of injection molding is more suitable for future commercial applications.

New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear

Characterizing purely viscous or purely elastic rheological nonlinearities is straightforward using rheometric tests such as steady shear or step strains. However, a definitive framework does not exist to characterize materials which exhibit both viscous and elastic nonlinearities simultaneously. We define a robust and physically meaningful scheme to quantify such behavior, using an imposed large amplitude oscillatory shear (LAOS) strain. Our new framework includes new material measures and clearly defined terminology such as intra-/intercycle nonlinearities, strain-stiffening/softening, and shear-thinning/thickening. The method naturally lends a physical interpretation to the higher Fourier coefficients that are commonly reported to describe the nonlinear stress response. These nonlinear viscoelastic properties can be used to provide a “rheological fingerprint” in a Pipkin diagram that characterizes the material response as a function of both imposed frequency and strain amplitude. We illustrate our new framework by first examining prototypical nonlinear constitutive models (including purely elastic and purely viscous models, and the nonlinear viscoelastic constitutive equation proposed by Giesekus). In addition, we use this new framework to study experimentally two representative nonlinear soft materials, a biopolymer hydrogel and a wormlike micelle solution. These new material measures can be used to characterize the rheology of any complex fluid or soft solid and clearly reveal important nonlinear material properties which are typically obscured by conventional test protocols.

On the rest state stability of an objective fractional derivative viscoelastic fluid model

The one-dimensional fractional derivative Maxwell model has been found very useful in modeling the linear viscoelastic response in the glass transition and α-relaxation regions. That motivated further work on generalizations to nonlinear viscoelastic fractional constitutive equations. In the work of Palade et al. [Int. J. Eng. Sci. 37, 315 (1999)], a fully objective constitutive equation for an incompressible fluid—reducible to the linear fractional derivative Maxwell model under the small deformations hypothesis—is given, together with a state of rest stability analysis. In the stability section of the aforementioned work, for some physically aceptable perturbation inputs, it has been shown that no continuously differentiable solutions may be found to the corresponding equations of motion. In this work, we extend the stability analysis framework to L2 Hilbert functional space and prove the existence of solutions to the corresponding equations of motions.

A Multi-Scale Finite Element Crash Analysis for Head Injury Prediction by Employing a Micro Blood Vessel Model

Head injury caused by the traffic crash accident is recognized as a serious problem for the establishment of the safety society. Many studies have been done to elucidate the injury generation mechanism of the brain by using the finite element (FE) analyses. In this study, a multi-scale finite element code is newly developed to assess the head injury by employing the stress and strain based criterion. The nonlinear viscoelastic constitutive equation is employed both for the macroscopic and the microscopic scales. Macroscopic head model consists of the skull, cerebral spinal fluid (CSF), facial bone and brain tissue. Microscopic model consists of the blood vessels, blood and brain tissue. When the human head is loaded by the translational acceleration in the traffic accident, the human brain is damaged at the crash side and the counter side, such as the coup and the contre-coup brain injuries. At the macro scale, the compressive hydraulic pressure was concentrated on the coup side, and the expansive hydraulic pressure on the contre-coup side. By employing the boundary velocity, which is derived from the macro FE analysis, the micro FE analysis is carried out. It shows that a very high equivalent stress, obtained by high tensile and shear stresses, occurred in the blood vessel. It was suggested that this multi-scale FE analysis could predict the blood vessel injury by employing the critical injury stress-strain criterion at the macro and micro scale.