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  • 1.
    Fredriksson, Per
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    A strain gradient plasticity analysis of size effects in thin films2005Licentiate thesis, comprehensive summary (Other scientific)
  • 2.
    Fredriksson, Per
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.), Solid Mechanics (Div.).
    Modelling and simulation of plastic deformation on small scales: interface conditions and size effects of thin films2008Doctoral thesis, comprehensive summary (Other scientific)
    Abstract [en]

    Contrary to elastic deformation, plastic deformation of crystalline materials, such as metals, is size-dependent. Most commonly, this phenomenon is present but unnoticed, such as the effect of microstructural length scales. The grain size in metallic materials is a length scale that affects material parameters such as yield stress and hardening moduli. In addition, several experiments performed in recent years on specimens with geometrical dimensions on the micron scale have shown that these dimensions also influence the mechanical behaviour. The work presented in this thesis involves continuum modelling and simulation of size-dependent plastic deformation, with emphasis on thin films and the formulation of interface conditions.

    A recently published strain gradient plasticity framework for isotropic materials [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379-1406] is used as a basis for the work. The theory is higher-order in the sense that additional boundary conditions are required and, as a consequence, higher-order stresses appear in the theory. For dimensional consistency, length scale parameters enter the theory, which is not the case for conventional plasticity theory. In Paper A and B, interface conditions are formulated in terms of a surface energy. The surface energy is assumed to depend on the plastic strain state at the interface and different functional forms are investigated. Numerical results are generated with the finite element method and it is found that this type of interface condition can capture the boundary layers that develop at the substrate interface in thin films. Size-effects are captured in the hardening behaviour as well as the yield strength. In addition, it is shown that there is an equivalence between a surface energy varying linearly in plastic strain and a viscoplastic interface law for monotonous loading.

    In paper C, a framework of finite element equations is formulated, of which a plane strain version is implemented in a commercial finite element program. Results are presented for an idealized problem of a metal matrix composite and several element types are examined numerically. In paper D, the implementation is used in a numerical study of wedge indentation of a thin film on an elastic substrate. Several trends that have been observed experimentally are captured in the theoretical predictions. Increased hardness at shallow depths due to gradient effects as well as increased hardness at more significant depths due to the presence of the substrate are found. It is shown that the hardening behaviour of the film has a large impact on the substrate effect and that either pile-up or sink-in deformation modes may be obtained depending on the material length scale parameter. Finally, it is qualitatively demonstrated that the substrate compliance has a significant effect on the calculated hardness of the film.

  • 3.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Gudmundson, Peter
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Competition between interface and bulk dominated plastic deformation in strain gradient plasticity2007In: Modelling and Simulation in Materials Science and Engineering, ISSN 0965-0393, E-ISSN 1361-651X, Vol. 15, no 1, p. S61-S69Article in journal (Refereed)
    Abstract [en]

    In the present report, the competition between dissipative plastic strain gradient effects in the bulk and in an interface is investigated within a strain gradient plasticity framework. The model of the interface is analysed in terms of hardening behaviour and strength for the case of a thin film with an elastic plastic interface. It is found that the yield strength of the film is increased by length scale effects both in the bulk material and the interface. The effect is governed by quite a simple rule, namely the weakest link of bulk and interface. In addition, if the interface is allowed to harden, three regions are observed for the bulk (interior) and interface of the film during an increasing load: (i) elastic bulk and rigid interface, (ii) both bulk and interface plastic and (iii) plastic bulk and rigid interface. The properties of the model are illustrated with numerical results from a parametric study.

  • 4.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Gudmundson, Peter
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Modelling of the interface between a thin film and a substrate within a strain gradient plasticity framework2007In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 55, no 5, p. 939-955Article in journal (Refereed)
    Abstract [en]

    Interfaces play an important role for the plastic deformation at the micron scale. In this paper, two types of interface models for isotropic materials are developed and applied in a thin film analysis. The first type, which can also be motivated from dislocation theory, assumes that the plastic work at the interface is stored as a surface energy that is linear in plastic strain. In the second model, the plastic work is completely dissipated and there is no build-up of a surface energy. Both formulations introduce one length scale parameter for the bulk material and one for the interface, which together control the film behaviour. It is demonstrated that the two interface models give equivalent results for a monotonous, increasing load. The combined influence of bulk and interface is numerically studied and it is shown that size effects are obtained, which are controlled by the length scale parameters of bulk and interface.

  • 5.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Gudmundson, Peter
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Size-dependent yield strength and surface energies of thin films2005In: Materials Science & Engineering: A, ISSN 0921-5093, E-ISSN 1873-4936, Vol. 400-401, no 1-2 SUPPL., p. 448-450Article in journal (Refereed)
    Abstract [en]

    The strain gradient plasticity theory recently proposed by Gudmundson [P. Gudmundson, J. Mech. Phys. Solids 52 (2004) 1379-1406] is used to analyse the behaviour of a thin film on an elastic substrate. Boundary conditions for the film-substrate interface are introduced via a surface energy that depends on the plastic strain state at the interface. Finite element results show a strong dependence on the surface energy. If the surface energy is small, no size effects appear. On the other hand, if a stiff interface is simulated, corresponding to a large surface energy, a thickness dependence of the yield strength is found. The application of several alternative strain gradient models would predict a thickness dependent hardening, but strictly not a size dependence of the yield strength. The presently predicted thickness dependence on yield strength and hardening is supported by experimental results.

  • 6.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Gudmundson, Peter
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Size-dependent yield strength of thin films2005In: International journal of plasticity, ISSN 0749-6419, E-ISSN 1879-2154, Vol. 21, no 9, p. 1834-1854Article in journal (Refereed)
    Abstract [en]

    Biaxial strain and pure shear of a thin film are analysed using a strain gradient plasticity theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379-1406]. Constitutive equations are formulated based on the assumption that the free energy only depends on the elastic strain and that the dissipation is influenced by the plastic strain gradients. The three material length scale parameters controlling the gradient effects in a general case are here represented by a single one. Boundary conditions for plastic strains are formulated in terms of a surface energy that represents dislocation buildup at an elastic/plastic interface. This implies constrained plastic flow at the interface and it enables the simulation of interfaces with different constitutive properties. The surface energy is also controlled by a single length scale parameter, which together with the material length scale defines a particular material. Numerical results reveal that a boundary layer is developed in the film for both biaxial and shear loading, giving rise to size effects. The size effects are strongly connected to the buildup of surface energy at the interface. If the interface length scale is small, the size effect vanishes. For a stiffer interface, corresponding to a non-vanishing surface energy at the interface, the yield strength is found to scale with the inverse of film thickness.

  • 7.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Gudmundson, Peter
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Mikkelsen, Lars Pilgaard
    Tech Univ Denmark, Risø Natl Lab Sustainable Energy, Mat Res Dept.
    Finite element implementation and numerical issues of strain gradient plasticity with application to metal matrix composites2009In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 46, no 22-23, p. 3977-3987Article in journal (Refereed)
    Abstract [en]

    A framework of finite element equations for strain gradient plasticity is presented. The theoretical framework requires plastic strain degrees of freedom in addition to displacements and a plane strain version is implemented into a commercial finite element code. A couple of different elements of quadrilateral type are examined and a few numerical issues are addressed related to these elements as well as to strain gradient plasticity theories in general. Numerical results are presented for an idealized cell model of a metal matrix composite under shear loading. It is shown that strengthening due to fiber size is captured but strengthening due to fiber shape is not. A few modelling aspects of this problem are discussed as well. An analytic solution is also presented which illustrates similarities to other theories.

  • 8.
    Fredriksson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Larsson, Per-Lennart
    KTH, School of Engineering Sciences (SCI), Solid Mechanics (Dept.).
    Wedge indentation of thin films modelled by strain gradient plasticity2008In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 45, no 21, p. 5556-5566Article in journal (Refereed)
    Abstract [en]

    A plane strain study of wedge indentation of a thin film on a substrate is performed. The film is modelled with the strain gradient plasticity theory by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. journal of the Mechanics and Physics of Solids 52, 1379-1406] and analysed using finite element simulations. Several trends that have been experimentally observed elsewhere are captured in the predictions of the mechanical behaviour of the thin film. Such trends include increased hardness at shallow depths due to gradient effects as well as increased hardness at larger depths due to the influence of the substrate. In between, a plateau is found which is observed to scale linearly with the material length scale parameter. It is shown that the degree of hardening of the material has a strong influence on the substrate effect, where a high hardening modulus gives a larger impact on this effect. Furthermore, pile-up deformation dominated by plasticity at small values of the internal length scale parameter is turned into sink-in deformation where plasticity is suppressed for larger values of the length scale parameter. Finally, it is demonstrated that the effect of substrate compliance has a significant effect on the hardness predictions if the effective stiffness of the substrate is of the same order as the stiffness of the film.

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