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  • 1.
    Elmukashf, Elsiddig
    et al.
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Numerical analysis of dynamic crack propagation in rubber2012Inngår i: International Journal of Fracture, ISSN 0376-9429, E-ISSN 1573-2673, Vol. 177, nr 2, s. 163-178Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.

  • 2.
    Elmukashfi, Elsiddig
    et al.
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Numerical analysis of dynamic crack propagation in biaxially strained rubber sheets2014Inngår i: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 124/125, s. 1-17Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper proposes a computational framework for dynamic crack propagation in rubber in which a nonlinear finite element analysis using cohesive zone modeling approach is used. A suddenly initiated crack at the center of biaxially stretched sheet problem is studied under plane stress conditions. A transient dynamic analysis using implicit time integration scheme is performed. In the constitutive modeling, the continuum is characterized by finite-viscoelasticity theory and coupled with the fracture processes using a cohesive zone model. This computational framework was introduced previously by the present authors (Elmukashfi and Kroon, 2012). In the current work, the use of a rate-dependent cohesive model is examined in addition to investigation of generalized biaxial loading cases. A Kelvin–Voigt element is used to describe the rate-dependent cohesive model wherein the spring is described by a bilinear law and dashpot with a constant viscosity is adopted. An explicit integration is used to incorporate the rate-dependent cohesive model in the finite element environment. A parametric study over the cohesive viscosity is performed and the steady crack propagation velocity is evaluated and compared with experimental data. It appears that the viscosity varies with the crack speed. Further, the total work of fracture is estimated using rate-independent cohesive law such that the strength of the cohesive zone is assumed to be constant and the separation work per unit area is determined form the experimental data. The results show that fracture-related processes, i.e. creation of new surfaces, cavitation and crystallization; contribute to the total work of fracture in a contradictory manner.

  • 3.
    Eriksson, Thomas
    et al.
    Graz University of Technology, Austria ; Royal Institute of Technology (KTH).
    Kroon, Martin
    Royal Institute of Technology (KTH).
    Holzapfel, Gerhard A.
    Graz University of Technology, Austria ; Royal Institute of Technology (KTH).
    Influence of medial collagen organization and in-situ axial stretch on saccular cerebral aneurysm growth2009Inngår i: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 131, nr 10, artikkel-id 101010Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A model for saccular cerebral aneurysm growth, proposed by Kroon and Holzapfel (2007, "A Model for Saccular Cerebral Aneurysm Growth in a Human Middle Cerebral Artery," J. Theor. Biol., 247, pp. 775-787; 2008, "Modeling of Saccular Aneurysm Growth in a Human Middle Cerebral Artery," ASME J. Biomech. Eng., 130, p. 051012), is further investigated. A human middle cerebral artery is modeled as a two-layer cylinder where the layers correspond to the media and the adventitia. The immediate loss of media in the location of the aneurysm is taken to be responsible for the initiation of the aneurysm growth. The aneurysmis regarded as a development of the adventitia, which is composed of several distinct layers of collagen fibers perfectly aligned in specified directions. The collagen fibers are the only load-bearing constituent in the aneurysm wall; their production and degradation depend on the stretch of the wall and are responsible for the aneurysm growth. The anisotropy of the surrounding media was modeled using the strain-energy function proposed by Holzapfel et al. (2000, "A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models," J. Elast., 61, pp. 1-48), which is valid for an elastic material with two families of fibers. It was shown that the inclusion of fibers in the media reduced the maximum principal Cauchy stress and the maximum shear stress in the aneurysm wall. The thickness increase in the aneurysm wall due to material growth was also decreased. Varying the fiber angle in the media from a circumferential direction to a deviation of 10 deg from the circumferential direction did, however, only show a little effect. Altering the axial in situ stretch of the artery had a much larger effect in terms of the steady-state shape of the aneurysm and the resulting stresses in the aneurysm wall. The peak values of the maximum principal stress and the thickness increase both became significantly higher for larger axial stretches.

  • 4.
    Fallqvist, B.
    et al.
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    A chemo-mechanical constitutive model for transiently cross-linked actin networks and a theoretical assessment of their viscoelastic behaviour2013Inngår i: Biomechanics and Modeling in Mechanobiology, ISSN 1617-7959, E-ISSN 1617-7940, Vol. 12, nr 2, s. 373-382Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Biological materials can undergo large deformations and also show viscoelastic behaviour. One such material is the network of actin filaments found in biological cells, giving the cell much of its mechanical stiffness. A theory for predicting the relaxation behaviour of actin networks crosslinked with the cross-linker α-actinin is proposed. The constitutive model is based on a continuum approach involving a neo-Hookean material model, modified in terms of concentration of chemically activated cross-links. The chemical model builds on work done by Spiros (Doctoral thesis, University of British Columbia, Vancouver, Canada, 1998) and has been modified to respond to mechanical stress experienced by the network. The deformation is split into a viscous and elastic part, and a thermodynamically motivated rate equation is assigned for the evolution of viscous deformation. The model predictions were evaluated for stress relaxation tests at different levels of strain and found to be in good agreement with experimental results for actin networks cross-linked with α-actinin.

  • 5.
    Fallqvist, B.
    et al.
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Constitutive modelling of composite biopolymer networks2016Inngår i: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 395, s. 51-61Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The mechanical behaviour of biopolymer networks is to a large extent determined at a microstructural level where the characteristics of individual filaments and the interactions between them determine the response at a macroscopic level. Phenomena such as viscoelasticity and strain-hardening followed by strain-softening are observed experimentally in these networks, often due to microstructural changes (such as filament sliding, rupture and cross-link debonding). Further, composite structures can also be formed with vastly different mechanical properties as compared to the individual networks. In this present paper, we present a constitutive model presented in a continuum framework aimed at capturing these effects. Special care is taken to formulate thermodynamically consistent evolution laws for dissipative effects. This model, incorporating possible anisotropic network properties, is based on a strain energy function, split into an isochoric and a volumetric part. Generalisation to three dimensions is performed by numerical integration over the unit sphere. Model predictions indicate that the constitutive model is well able to predict the elastic and viscoelastic response of biological networks, and to an extent also composite structures.

  • 6.
    Fallqvist, B.
    et al.
    Royal Institute of Technology, (KTH).
    Kulachenko, A.
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Modelling of cross-linked actin networks: influence of network parameters and cross-link compliance2014Inngår i: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 350, s. 57-69Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A major structural component of the cell is the actin cytoskeleton, in which actin subunits are polymerised into actin filaments. These networks can be cross-linked by various types of ABPs (Actin Binding Proteins), such as Filamin A. In this paper, the passive response of cross-linked actin filament networks is evaluated, by use of a numerical and continuum network model. For the numerical model, the influence of filament length, statistical dispersion, cross-link compliance (including that representative of Filamin A) and boundary conditions on the mechanical response is evaluated and compared to experimental results. It is found that the introduction of statistical dispersion of filament lengths has a significant influence on the computed results, reducing the network stiffness by several orders of magnitude. Actin networks have previously been shown to have a characteristic transition from an initial bending-dominated to a stretching-dominated regime at larger strains, and the cross-link compliance is shown to shift this transition. The continuum network model, a modified eight-chain polymer model, is evaluated and shown to predict experimental results reasonably well, although a single set of parameters cannot be found to predict the characteristic dependence of filament length for different types of cross-links. Given the vast diversity of cross-linking proteins, the dependence of mechanical response on cross-link compliance signifies the importance of incorporating it properly in models to understand the roles of different types of actin networks and their respective tasks in the cell.

  • 7.
    Fallqvist, Björn
    et al.
    Royal Institute of Technology, (KTH).
    Fielden, Matthew L.
    Royal Institute of Technology, (KTH).
    Pettersson, Torbjörn
    Royal Institute of Technology, (KTH).
    Nordgren, Niklas
    SP Technical Research Institute of Sweden.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Gad, Annica K.B.
    Karolinska Institutet, KI.
    Experimental and computational asessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts2016Inngår i: Journal of The Mechanical Behavior of Biomedical Materials, ISSN 1751-6161, E-ISSN 1878-0180, Vol. 59, s. 168-184Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In biomechanics, a complete understanding of the structures and mechanisms that regulate cellular stiffness at a molecular level remain elusive. In this paper, we have elucidated the role of filamentous actin (F-actin) in regulating elastic and viscous properties of the cytoplasm and the nucleus. Specifically, we performed colloidal-probe atomic force microscopy (AFM) on BjhTERT fibroblast cells incubated with Latrunculin B (LatB), which results in depolymerisation of F-actin, or DMSO control. We found that the treatment with LatB not only reduced cellular stiffness, but also greatly increased the relaxation rate for the cytoplasm in the peripheral region and in the vicinity of the nucleus. We thus conclude that F-actin is a major determinant in not only providing elastic stiffness to the cell, but also in regulating its viscous behaviour. To further investigate the interdependence of different cytoskeletal networks and cell shape, we provided a computational model in a finite element framework. The computational model is based on a split strain energy function of separate cellular constituents, here assumed to be cytoskeletal components, for which a composite strain energy function was defined. We found a significant influence of cell geometry on the predicted mechanical response. Importantly, the relaxation behaviour of the cell can be characterised by a material model with two time constants that have previously been found to predict mechanical behaviour of actin and intermediate filament networks. By merely tuning two effective stiffness parameters, the model predicts experimental results in cells with a partly depolymerised actin cytoskeleton as well as in untreated control. This indicates that actin and intermediate filament networks are instrumental in providing elastic stiffness in response to applied forces, as well as governing the relaxation behaviour over shorter and longer time-scales, respectively.

  • 8.
    Fallqvist, Björn
    et al.
    Royal Institute of Technology, (KTH).
    Kulachenko, Artem
    Royal Institute of Technology, (KTH).
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Cross-link debonding in actin networks: influence on mechanical properties2015Inngår i: International Journal of Experimental and Computational Biomechanics, ISSN 1755-8735, Vol. 3, nr 1, s. 16-26Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The actin cytoskeleton is essential for the continued function and survival of the cell. A peculiar mechanical characteristic of actin networks is their remodelling ability, providing them with a time-dependent response to mechanical forces. In cross-linked actin networks, this behaviour is typically tuned by the binding affinity of the cross-link. We propose that the debonding of a cross-link between filaments can be modelled using a stochastic approach, in which the activation energy for a bond is modified by a term to account for mechanical strain energy. By use of a finite element model, we perform numerical analyses in which we first compare the model behaviour to experimental results. The computed and experimental results are in good agreement for short time scales, but over longer time scales the stress is overestimated. However, it does provide a possible explanation for experimentally observed strain-rate dependence as well as strain-softening at longer time scales.

  • 9.
    Holzapfel, Gerhard A.
    et al.
    Royal Institute of Technology KTH ; Graz University of Technology, Austria.
    Kiousis, Dimitrios E.
    Graz University of Technology, Austria .
    Kroon, Martin
    Royal Institute of Technology KTH.
    On Modelling Multi-Layered Soft Collagenous Tissues2008Inngår i: Presented at 8th World Congress on Computational Mechanics, 30 June – 4 July, 2008, Venice, Italy, 2008Konferansepaper (Fagfellevurdert)
  • 10.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    A constitutive framework for modelling thin incompressible viscoelastic materials under plane stress in the finite strain regime2011Inngår i: Mechanics of time-dependant materials, ISSN 1385-2000, E-ISSN 1573-2738, Vol. 15, nr 4, s. 389-406Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Rubbers and soft biological tissues may undergo large deformations and are also viscoelastic. The formulation of constitutive models for these materials poses special challenges. In several applications, especially in biomechanics, these materials are also relatively thin, implying that in-plane stresses dominate and that plane stress may therefore be assumed. In the present paper, a constitutive model for viscoelastic materials in the finite strain regime and under the assumption of plane stress is proposed. It is assumed that the relaxation behaviour in the direction of plane stress can be treated separately, which makes it possible to formulate evolution laws for the plastic strains on explicit form at the same time as incompressibility is fulfilled. Experimental results from biomechanics (dynamic inflation of dog aorta) and rubber mechanics (biaxial stretching of rubber sheets) were used to assess the proposed model. The assessment clearly indicates that the model is fully able to predict the experimental outcome for these types of material.

  • 11.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    A constitutive model for smooth muscle including active tone and passive viscoelastic behaviour2010Inngår i: Mathematical Medicine and Biology, ISSN 1477-8599, E-ISSN 1477-8602, Vol. 27, nr 2, s. 129-155Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The model includes a chemical part, governing the cross-bridge (myosin head) cycling, that is responsible for the filament sliding. The number of activated cross-bridges govern the contractile force generated and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive viscoelastic part. The constitutive model was calibrated with respect to experiments on smooth muscle tissue from swine carotid artery and guinea pig taenia coli, in terms of isometric and isotonic tensile test results. The model was fully able to reproduce the experimental results.

  • 12.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    A constitutive model for strain-crystallising rubber-like materials2010Inngår i: Mechanics of materials (Print), ISSN 0167-6636, E-ISSN 1872-7743, Vol. 42, nr 9, s. 873-885Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In the present paper, a constitutive model for strain-crystallising rubber is proposed. The constitutive behaviour is formulated in terms of a strain energy function, where the full network approach is adopted. The Arrhenius equation provides the basis for the crystallite nucleation law. The full network approach allows for the development of an anisotropic crystal structure. The model was applied to experimental results from uniaxial tensile tests. Strain-crystallisation causes a hysteresis in the stress–stretch relation, but according to the model predictions, the effect of crystallisation is not sufficient to explain the mechanical hysteresis observed in the tensile tests. Hence, additional viscoelasticity associated with amorphous polymer chains must be included. The model was fully able to predict both the stress vs. stretch relations and the crystallinity vs. stretch relations from the experiments.

  • 13.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    A continuum mechanics framework and a constitutive model for remodelling of collagen gels and collagenous tissues2010Inngår i: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 58, nr 6, s. 918-933Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Collagen is a very important protein of the human body and is responsible for the structural stability of many body components. Furthermore, collagen fibre networks are able to grow and remodel themselves, which enables them to adjust to varying physiological conditions. This remodelling is accomplished by fibre-producing cells, such as fibroblasts. The ability to adjust to new physiological conditions is very important, for example in wound healing. In the present paper, a theoretical framework for modelling collagenous tissues and collagen gels is proposed. Continuum mechanics is employed to describe the kinematics of the collagen, and affine deformations of fibres are assumed. Biological soft tissues can be approximated as being hyperelastic, and the constitutive model for the collagen fabric is therefore formulated in terms of a strain energy function. This strain energy function includes a density function that describes the distribution of the collagen fibre orientation. The density function evolves according to an evolution law, where fibres tend to reorient towards the direction of maximum Cauchy stress. The remodelling of the collagen network is also assumed to include a pre-stretching of collagen fibres, accomplished by fibroblasts. The theoretical framework is applied to experiments performed on collagen gels, where gels were exposed to remodelling under both biaxial and uniaxial constraints. The proposed model was able to predict both the resulting collagen distribution and the resulting stress–strain relationships obtained for the remodelled collagen gels. The influence of the most important model parameters is demonstrated, and it appears that there is a fairly unique set of model parameters that gives an optimal fit to the experimental data.

  • 14.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    A numerical framework for material characterization of inhomogeneous hyperelastic membranes by inverse analysis2010Inngår i: Journal of Computational and Applied Mechanics, ISSN 1586-2070, Vol. 234, nr 2, s. 563-578Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    An inverse method for estimating the distributions of the elastic properties of hyperelastic, inhomogeneous membranes is proposed. The material description of the membrane is based on a versatile constitutive model, in which two stiffness parameters govern the nonlinear elastic behaviour of the material. The estimation procedure includes a finite element framework. The two stiffness parameters in the constitutive law are assumed to vary continuously over the inhomogeneous membrane, and in the finite element framework the distributions of the two parameters are approximated using standard linear shape functions. Experimental results are assumed to exist in terms of nodal displacements from a test with known geometry and boundary conditions. The experimental membrane is modelled in the finite element framework, and the deformation of it is predicted. An error function, quantifying the discrepancy between the experimentally obtained deformation pattern and the numerically predicted pattern, is then minimised with respect to the nodal values of the two interpolated parameters. In a number of numerical examples, the proposed procedure is assessed by attempting to reproduce the given random reference distributions of material properties. The proposed estimation method is fully able to reproduce the reference distributions of the two material parameters with excellent accuracy.

  • 15. Kroon, Martin
    A Probabilistic Model for Cleavage Fracture with a Length Scale2003Inngår i: Presented at 5th Euromech Solid Mechanics Conference, 17-22 August, 2003, Thessaloniki, Greece, 2003Konferansepaper (Fagfellevurdert)
  • 16.
    Kroon, Martin
    Royal Institute of Technology KTH.
    A theoretical assessment of the influence of myosin filament dispersion on smooth muscle contraction2011Inngår i: Presented at ASME Summer Bioengineering Conference, 22-25 June, 2011, ASME Press, 2011, s. 145-146Konferansepaper (Fagfellevurdert)
    Abstract [en]

    A new constitutive model for the biomechanical behavior of smooth muscle tissue is employed to investigate the influence of statistical dispersion in the orientation of myosin filaments. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behavior of the smooth muscle tissue. The predictions from the constitutive model are compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress for a case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch.

  • 17. Kroon, Martin
    A Theoretical Model for Saccular Cerebral Aneurysm Growth: Deformation and Stress Analysis2007Inngår i: Presented at Summer Bioengineering Conference, 20-24 June, 2007, Keystone, Colorado, 2007Konferansepaper (Fagfellevurdert)
  • 18.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    An 8-chain model for rubber-like materials accounting for non-affine chain deformations and topological constraints2011Inngår i: Journal of elasticity, ISSN 0374-3535, E-ISSN 1573-2681, Vol. 102, nr 2, s. 99-116Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Several industrial applications involve rubber and rubber-like materials, and it is important to be able to predict the constitutive response of these materials. In the present paper, a new constitutive model for rubber-like solids is proposed. The model is based on the 8-chain concept introduced by Arruda and Boyce (J. Mech. Phys. Solids 41, 389–412, 1993) to which two new components are added. Real polymer networks do not deform affinely, and in the proposed model this is accounted for by the inclusion of an elastic spring, acting in series with the representative polymer chain. Furthermore, real polymer chains are not completely free to move, which is modelled by imposing a topological constraint on the transverse motions of the representative polymer chain. The model contains five model parameters and these need to be determined on the basis of experimental data. Three experimental studies from the literature were used to assess the proposed model. The model was able to reproduce experimental data performed under conditions of uniaxial tension, generalised plane deformation, and biaxial tension with an excellent accuracy. The strong predictive abilities together with the numerically efficient structure of the model make it suitable for implementation in a finite element context.

  • 19.
    Kroon, Martin
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för maskinteknik (MT).
    An Anisotropic Lagrangian Plasticity Model for Semi-crystalline Polymers at Finite Strains2017Inngår i: US National Congress on Computational Mechanics, Montreal, July 17-20, 2017, 2017Konferansepaper (Fagfellevurdert)
    Abstract [en]

    The present work concerns modelling of semi-crystalline polymers, such as polyethylene. Semi-crystalline polymers constitute a material group with an increasing industrial importance. Constitutive response (stress-strain relation) as well as fracture properties are of great importance for this class of materials. We propose an anisotropic plasticity model for semi-crystalline polymers, formulated in the reference configuration (Lagrangian formulation). In addition to plasticity, the model accounts for effects of viscoelasticity. The model is evaluated using uniaxial tensile tests, where polyethylene sheets are tested. The model is able to predict the hardening and rate-dependency observed in the experiments very well. Numerical examples are also presented, where the model is applied to a few different geometries

  • 20.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    An asymptotic analysis of dynamic crack growth in rubber2011Inngår i: Engineering Fracture Mechanics, ISSN 0013-7944, E-ISSN 1873-7315, Vol. 78, nr 17, s. 3111-3122Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks are important for several reasons. For example, they facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. The present study focuses on the asymptotic fields in front of a crack that propagates dynamically in rubber. Static analyses of this type of problem have been made in previous studies. In order to be able to compare the present results with these earlier studies, the constitutive model from Knowles and Sternberg (J. Elast. 3:67–107, 1973) was adopted. It is assumed that viscoelastic stresses become negligible compared with the singular elastic stresses close to the crack tip. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. However, for a neoHookean type of model, inertia comes into play and causes a maximum theoretical crack speed that equals the shear wave speed.

  • 21.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    An efficient method for material characterization of hyperelastic anisotropic inhomogeneous membranes based on inverse finite element method and an element partition strategy2010Inngår i: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 63, nr 2, s. 201-225Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    An inverse method for estimating the distributions of the nonlinear elastic properties of inhomogeneous and anisotropic membranes is investigated. The material description of the membrane is based on a versatile constitutive model, including four material parameters: two initial stiffness values pertaining to the principal directions of the material, the angle between these principal directions and a reference coordinate system and a parameter related to the level of nonlinearity of the material. The estimation procedure consists of the following three steps: (i) perform experiments on the membranous structure whose properties are to be determined, (ii) define a corresponding finite-element (FE) model and (iii) minimise an error function (with respect to the unknown parameters) that quantifies the deviation between the numerical predictions and the experimental data. For this finite deformation problem, an FE framework for membranous structures exposed to a pressure boundary loading is outlined: the principle of virtual work, its linearisation and the related spatial discretisation. To achieve a robust parameter estimation, an element partition method is employed. In numerical examples, the proposed procedure is assessed by attempting to reproduce given random reference distributions of material fields in a reference membrane. The deviations between the estimated material parameter distributions and the related reference fields are within a few percent in most cases. The standard deviation for the resulting maximum principal stress was consistently below 1%.

  • 22. Kroon, Martin
    An Inverse Method to Estimate the Material Parameters and Wall Stresses of a Saccular Cerebral Aneurysm2008Inngår i: Presented at 11th Euromech – Mecamat Conference, 10-14 March, 2008, Torino, Italy, 2008Konferansepaper (Fagfellevurdert)
  • 23. Kroon, Martin
    Analysis of dynamic crack propagation in rubber2012Inngår i: Presented at 10th World Congress on Computational Mechanics, 8-13 July, 2012, 2012Konferansepaper (Fagfellevurdert)
  • 24.
    Kroon, Martin
    Royal Institute of Technology KTH.
    Assessment of three possible criteria for remodelling of collagen gels and collagenous tissues2010Inngår i: Presented at ASME Summer Bioengineering Conference, 16-19 June, 2010, ASME Press, 2010, s. 775-776Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Collagenous tissues are living structures, in which new material may be added and the structural organisation may change over time. The maintenance of the collagen matrix is accomplished by fibre-producing cells, such as fibroblasts. During maintenance, the extracellular matrix (ECM) influences the development, shape, migration, proliferation, survival, and function of the cells. The mobility of the fibroblasts and their ability to contract the ECM are important properties for a proper maintenance of the ECM [1,2]. The purpose of the present paper is to shed some more light on the interaction between the ECM and the fibre-producing cells. The fibroblasts remodel the collagen gel by reorienting the individual collagen fibres. This reorientation of fibres is described by an evolution law, which depends on a continuum mechanics entity. Three possible choices are assessed: reorientation towards increasing Cauchy stress, increasing elastic stretch, and increasing current stiffness of the material. The model is compared with experimental results, and the three different criteria are evaluated in terms of the predicted distribution of collagen fibres after remodeling and resulting stress-strain relations. Experimental results from tissue equivalents in the form of collagen gels are used when assessing the three criteria [3]. We consider a network of collagen fibres, where the fibres are embedded in a matrix fluid. The collagen fabric and the surrounding fluid are assumed to be the only load-carrying constituents in the material. Embedded in and attached to the collagen fabric is also a population of fibroblasts. The collagen fabric is composed of collagen fibres, which in turn are bundles of collagen fibrils. The deformation of a line element in the matrix is described by the deformation gradient F(X) = ∂x/∂X, which is decomposed according to F = FelFlfFr, see Fig. 1. The fibroblasts’ remodelling of the collagen fabric results in a new matrix configuration Ωr. This deformation of the matrix is described by Fr. The configuration Ωr does not necessarily fulfill equilibrium, and the deformation gradient Flf takes the matrix to the state Ωlf, that fulfils global equilibrium with no external loads applied. Finally, if external loads are applied to the material, the configuration Ωel is attained, and this deformation is described by the deformation gradient Fel.

  • 25.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Asymptotic mechanical fields at the tip of a mode I crack in rubber-like solids2014Inngår i: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 51, nr 10, s. 1923-1930Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth and failure. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. In the present work, we perform an asymptotic analysis of the mechanical fields in the vicinity of a propagating mode I crack in rubber. Plane deformation is assumed, and the material model is based on the Langevin function, which accounts for the finite extensibility of polymer chains. The Langevin function is approximated by a polynomial, and only the term of the highest order contributes to the asymptotic solution. The crack is predicted to adopt a wedge-like shape, i.e. the crack faces will be straight lines. The angle of the wedge and the order of the stress singularity depend on the hardening of the strain energy function. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. Hence, there is no upper theoretical limit to the crack speed.

  • 26.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Cell contraction of an elastic substrate assessed by an axisymmetric model2012Inngår i: International Journal of Experimental and Computational Biomechanics, ISSN 1755-8735, Vol. 2, nr 1, s. 61-73Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In the present paper, a computational model for cell contraction of an elastic substrate is proposed. Axisymmetry is assumed and the cell is represented by its contractile apparatus, which is taken to consist of radially oriented stress fibres. The constitutive behaviour of the contractile apparatus is modelled by use of a strain energy function, and contraction of stress fibres is modelled by giving them a different natural configuration compared with the underlying elastic substrate. The model was compared with experiments, in which fibroblasts were put on an elastic substrate. The contracted cell radius depends on the stiffness of the elastic substrate, and model predictions were compared with the experimental results for different values of the stiffness of the elastic substrate. The model also predicts that the contraction of the cell tends to cause a small crater below the cell, which is qualitatively in agreement with experimental observations.

  • 27.
    Kroon, Martin
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för maskinteknik (MT).
    Crack Growth in Low-density Polyethylene2018Inngår i: Presented at EMMC16, European Mechanics of Materials Conference in Nantes, France, 26-28 March, 2018, 2018Konferansepaper (Fagfellevurdert)
  • 28. Kroon, Martin
    Dispersion Effects of Active Contractile Filaments in Smooth Muscle Cells2009Inngår i: Presented at 4th International Congress on Computational Bioengineering, 16-18 September, 2009, 2009Konferansepaper (Fagfellevurdert)
  • 29.
    Kroon, Martin
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för maskinteknik (MT).
    Dynamic Crack propagation in Rubber2016Inngår i: Presented at 24th International Congress of Theoretical and Applied Mechanics (ICTAM), Montreal, Canada, August 21-26, 2016, 2016Konferansepaper (Fagfellevurdert)
  • 30.
    Kroon, Martin
    Malmö University.
    Dynamic Energy Release Rates in Rubber2016Inngår i: Presented at European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Crete, Greece, June 5-10, 2016, 2016Konferansepaper (Fagfellevurdert)
  • 31.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Dynamic steady-state analysis of crack propagation in rubber-like solids using an extended finite element method2012Inngår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 49, nr 1, s. 73-86Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In the present study, a computational framework for studying high-speed crack growth in rubber-like solids under conditions of plane stress and steady-state is proposed. Effects of inertia, viscoelasticity and finite strains are included. The main purpose of the study is to examine the contribution of viscoelastic dissipation to the total work of fracture required to propagate a crack in a rubber-like solid. The computational framework builds upon a previous work by the present author (Kroon in Int J Fract 169:49-60, ). The model was fully able to predict experimental results in terms of the local surface energy at the crack tip and the total energy release rate at different crack speeds. The predicted distributions of stress and dissipation around the propagating crack tip are presented. The predicted crack tip profiles also agree qualitatively with experimental findings.

  • 32. Kroon, Martin
    Dynamic steady-state crack propagation in rubber-like solids2011Konferansepaper (Fagfellevurdert)
  • 33.
    Kroon, Martin
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för maskinteknik (MT).
    Energy release rates in rubber during dynamic crack propagation2015Inngår i: Presented at ASME Applied Mechanics and Material Conference, July 25-29, 2015, 2015Konferansepaper (Fagfellevurdert)
  • 34.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Energy release rates in rubber during dynamic crack propagation2014Inngår i: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 51, nr 25-26, s. 4419-4426Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The theoretical understanding of the fracture mechanics of rubber is not as well developed as for other engineering materials, such as metals. The present study is intended to further the understanding of the dissipative processes that take place in rubber in the vicinity of a propagating crack tip. This dissipation contributes significantly to the total fracture toughness of the rubber and is therefore of great interest from a fracture mechanics point of view. To study this, a computational framework for analysing high-speed crack growth in a biaxially stretched rubber under plane stress is therefore formulated. The main purpose is to investigate the energy release rates required for crack propagation under different modes of biaxial stretching. The results show, that inertia comes into play when the crack speed exceeds about 50 m/s. The total work of fracture by far exceeds the surface energy consumed at the very crack tip, and the difference must be attributed to dissipative damage processes in the vicinity of the crack tip. The size of this damage/dissipation zone is expected to be a few millimetres.

  • 35.
    Kroon, Martin
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för maskinteknik (MT).
    Experimental measurement of energy release rate in polyethylene2017Inngår i: The 14th International Conference on Fracture, Rhodes, Greece, June 18-23, 2017, Rhodes, 2017Konferansepaper (Fagfellevurdert)
  • 36.
    Kroon, Martin
    Royal Institute of Technology, (KTH).
    Influence of dispersion in myosin filament orientation and anisotropic filament contractions in smooth muscle2011Inngår i: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 272, nr 1, s. 72-82Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The orientation of the myosin filaments, and thereby the contractile units, are taken to exhibit a statistical dispersion around a preferred direction. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive elastic part. The constitutive model was compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress for a case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch.

  • 37. Kroon, Martin
    Material Characterization of Biological Membranes by Inverse Analysis2008Inngår i: Presented at 22nd International Congress of Theoretical and Applied Mechanics, 24-30 August, 2008, 2008Konferansepaper (Fagfellevurdert)
  • 38.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    Material Properties of Inhomogeneous Hyperelastic Membranes Assessed by Inverse Analysis2009Konferansepaper (Fagfellevurdert)
  • 39. Kroon, Martin
    Mechanical Modelling of Calcium-Activated Contraction of Smooth Muscle Cells2008Inngår i: Presented at IUTAM Symposium on Cellular, Molecular and Tissue Mechanics, 18-21 June, 2008, Woods Hole, Massachusetts, USA, 2008Konferansepaper (Fagfellevurdert)
  • 40.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    Micromechanical Analysis of Cleavage Fracture Initiation2003Inngår i: 9th International Conference on The Mechanical Behaviour of Materials, Palexpo congress center, 25-29 May, 2003, Geneva, 2003Konferansepaper (Fagfellevurdert)
  • 41.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    Modeling of fibroblast-controlled strengthening and remodeling of uniaxially constrained collagen gels2010Inngår i: Journal of Biomechanical Engineering, ISSN 0148-0731, E-ISSN 1528-8951, Vol. 132, nr 11, artikkel-id 111008Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A theoretical model for the remodeling of collagen gels is proposed. The collagen fabric is modeled as a network of collagen fibers, which in turn are composed of collagen fibrils. In the model, the strengthening of collagen fabric is accomplished by fibroblasts, which continuously recruit and attach more collagen fibrils to existing collagen fibers. The fibroblasts also accomplish a reorientation of collagen fibers. Fibroblasts are assumed to reorient collagen fibers toward the direction of maximum material stiffness. The proposed model is applied to experiments in which fibroblasts were inserted into a collagen gel. The model is able to predict the force-strain curves for the experimental collagen gels, and the final distribution of collagen fibers also agrees qualitatively with the experiments.

  • 42. Kroon, Martin
    Modelling of fibroblast-controlled restructuring of collagen gels2010Inngår i: Presented at 6th World Congress in Biomechanics, 1-6 August, 2010, 2010Konferansepaper (Fagfellevurdert)
  • 43. Kroon, Martin
    Modelling of smooth muscle Cells2008Inngår i: Presented at 8th World Congress on Computational Mechanics, 30 June – 4 July, 2008, Venice, Italy, 2008Konferansepaper (Fagfellevurdert)
  • 44. Kroon, Martin
    Modelling of Thin Anisotropic Collagen-Dominated Soft Biological Tissue2008Inngår i: Presented at 2nd International Conference on Heterogeneous Material Mechanics, 3-8 June, 2008, Huangshan, China, 2008Konferansepaper (Fagfellevurdert)
  • 45. Kroon, Martin
    Numerical analysis of steady-state crack growth in rubber2012Inngår i: Presented at 10th World Congress on Computational Mechanics, 8-13 July, 2012, 2012Konferansepaper (Fagfellevurdert)
  • 46. Kroon, Martin
    Numerical analysis of steady-state crack growth in rubber-like solids2011Inngår i: Presented at US National Congress on Computational Mechanics, 24-28 July, 2011, 2011Konferansepaper (Fagfellevurdert)
  • 47.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    On the correlation between continuum mechanics entities and cell activity in biological soft tissues: Assessment of three possible criteria for cell-controlled fibre reorientation in collagen gels and collagenous tissues2010Inngår i: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 264, nr 1, s. 66-76Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The biomechanical behaviour of biological cells is of great importance in many physiological processes. One such process is the maintenance of fibrous networks, such as collagenous tissues. The activity of the fibre-producing cells in this type of tissue is very important, and a comprehensive material description needs to incorporate the activity of the cells. In biomechanics, continuum mechanics is often employed to describe deforming solids, and modelling can be much simplified if continuum mechanics entities, such as stress and strain, can be correlated with cell activity. To investigate this, a continuum mechanics framework is employed in which remodelling of a collagen gel is modelled. The remodelling is accomplished by fibroblasts, and the activity of the fibroblasts is linked to the continuum mechanics theory. The constitutive model for the collagen fabric is formulated in terms of a strain energy function, which includes a density function describing the distribution of the collagen fibre orientation. This density function evolves according to an evolution law, where fibroblasts reorient fibres towards the direction of increasing Cauchy stress, elastic deformation, or stiffness. The theoretical framework is applied to experimental results from collagen gels, where gels have undergone remodelling under both biaxial and uniaxial constraint. The analyses indicated that criteria 1 and 2 (Cauchy stress and elastic deformations) are able to predict the collagen fibre distribution after remodelling, whereas criterion 3 (current stiffness) is not. This conclusion is, however, tentative and pertains, strictly speaking, only to fibre remodelling processes, and may not be valid for other types of cell activities.

  • 48.
    Kroon, Martin
    Royal Institute of Technology (KTH).
    Optimal length of smooth muscle assessed by a microstructurally and statistically based constitutive model2011Inngår i: Computer Methods in Biomechanics and Biomedical Engineering, ISSN 1025-5842, E-ISSN 1476-8259, Vol. 14, nr 1, s. 43-52Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Smooth muscle exhibits an optimal length at which it is able to generate a maximum amount of force. In this study, the optimal length is assessed by use of a microstructurally and statistically based constitutive model for smooth muscle. The model is based on the sliding filament theory, and a modified version of Hill's mechanical model was adopted. It was conjectured, that a variation in the overlap in the actomyosin contractile units together with a statistical dispersion in the size of the dense bodies are responsible for the optimal length characteristics. The influence of contractile unit length, dense body size and dense body compliance was investigated, and the model was fully able to predict experimental data. The results indicate that the compliance of the dense bodies does not contribute significantly to the total compliance of the contractile apparatus.

  • 49.
    Kroon, Martin
    Royal Institute of Technology.
    Probabilistic and micromechanical modelling of cleavage fracture2003Doktoravhandling, med artikler (Annet vitenskapelig)
  • 50. Kroon, Martin
    Saccular Aneurysm Growth in a Human Middle Cerebral Aneurysm: Deformation and Stress Analysis2007Inngår i: Presented at 44th Annual Meeting of the Society of Engineering Science, 21-24 October, 2007, 2007Konferansepaper (Fagfellevurdert)
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