ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Copyright © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Recently Published Articles
- Modelling and simulation of acrylic bone cement injection and curing within the framework of vertebroplasty
Ralf Landgraf, Jörn Ihlemann, Sebastian Kolmeder, Alexander Lion, Helena Lebsack and Cornelia Kober
Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400064
The minimal invasive procedure of vertebroplasty is a surgical technique to treat compression fractures of vertebral bodies. During the treatment, liquid bone cement gets injected into the affected vertebral body and therein cures to a solid. In order to investigate the treatment and the impact of injected bone cement, an integrated modelling and simulation framework has been developed. The framework includes (i) the generation of microstructural computer models based on microCT images of human cancellous bone, (ii) computational fluid dynamics (CFD) simulations of bone cement injection into the trabecular structure and (iii) non-linear finite element (FE) simulations of the subsequent bone cement curing. A detailed description of the material behaviour of acrylic bone cements is provided for both simulation stages. A non-linear process-dependent viscosity function is chosen to represent the bone cement behaviour during injection. The bone cements phase change from a highly viscous fluid to a solid is described by a non-linear viscoelastic material model with curing dependent properties. To take into account the distinctive temperature dependence of acrylic bone cements, both material models are formulated in a thermo-mechanically coupled manner. Moreover, the corresponding microstructural CFD- and FE-simulations are performed using thermo-mechanically coupled solvers. An application of the presented modelling and simulation framework to a sample of human cancellous bone demonstrates the capabilities of the presented approach.
- Existence and uniqueness for frictional incremental and rate problems – sharp critical bounds
L.-E. Andersson, A. Pinto da Costa and M. A. Agwa
Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400143
We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented.
- On a non-stationary load on the surface of a semiplane with mixed boundary conditions
Veniamin D. Kubenko
Article first published online: 26 JAN 2015 | DOI: 10.1002/zamm.201400202
An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the object at an arbitrary point of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.
- Classical solutions for a modified Hele-Shaw model with elasticity
Helmut Abels and Stefan Schaubeck
Article first published online: 22 JAN 2015 | DOI: 10.1002/zamm.201400099
For a large class of initial data, we prove the existence of classical solutions locally in time to a modified Hele-Shaw problem that takes elastic effects into account. The system arises as sharp interface model of a Cahn-Hilliard system coupled with linearized elasticity. By using the Hanzawa transformation, we can reduce the system to a single evolution equation for the height function. Then short time existence is proven by inverting the linearized operator and applying the contraction mapping principle.
- Analytical elastic-plastic analyses of a spherical shell subjected to hydrostatic tension based on a strain gradient model for plastic metals
Article first published online: 21 JAN 2015 | DOI: 10.1002/zamm.201400131
The problem of a spherical shell made of an elastic-plastic second gradient model for plastic materials and subjected to hydrostatic tension is considered. The elastic-plastic second gradient model is a simplified version (porosity neglected) of a second gradient model for plastic porous metals developed, some years ago, by Gologanu, Leblond, Perrin and Devaux, so-called GLPD model. The expressions of the velocity field as well as the ordinary and double stress components are determined for the cases where the spherical shell is modeled by a purely elastic, purely plastic, and elastic-plastic GLPD models. As expected, the solution developed in each case (elastic, ideal-plastic, and elastic-plastic) reduces to that of the first gradient as a special case when the characteristic length scale the GLPD model involves is negligible. Our results allow comparisons between the newly developed solution and the classical elastic-plastic solution for the same model problem; they also provide insights into the influence of the characteristic length scale on the newly developed solution.