SUMMARYThis paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains. Focus is on localized plastic deformation particularly relevant for slip bands. In contrast to already existing models, the proposed implementation allows to consider several interacting discontinuities in each finite element. Based on a proper re-formulation of the kinematics, an efficient parameterization of the deformation gradient is derived. It permits to compute the strains explicitly that improves the performance significantly. However, the most important novel contribution of the present paper is the advocated variational constitutive update. Within this framework, every aspect is naturally driven by energy minimization, i.e. all unknown variables are jointly computed by minimizing the stress power. The proposed update relies strongly on an extended principle of maximum dissipation. This framework provides enough flexibility for different failure types and for a broad class of non-associative evolution equations. By discretizing the aforementioned continuous variational principle, an efficient numerical implementation is obtained. It shows, in addition to its physical and mathematical elegance, several practical advantages. For instance, the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities.

Summary Introduction There is growing interest in the world for estimating the cost for the treatment of a disease. This value can be used to determine to which extent a particular disease or group of diseases burden society in terms of the global crisis (Segel 2006). In 2000, Organization for Economic Countries Development (OECD) established a System of Health Accounts (SHA), and provided methodological guide for calculating the cost of treating the disease. The aim of this study was to determine the cost of individual health care in the Republic of Serbia according to the major International Classification of Diseases (ICD) for the period 2010-2015. Material and Methods A retrospective and comparative analysis of health statistics from the database of the Institute of Public Health of Serbia (IPHS) and financial information provided by the National Health Insurance Fund (NHIF) in the period 2010-2015 was performed. Financial information and data on hospital services, outpatient, home health care, auxiliary health care services, drug consumption and consumer goods in healthcare were analyzed using SHA methodology. Results showed that during observation period the maximum cost of individual health care in Serbia by main classification ICD was achieved in 2015 and it was 194,128,864,011 RSD (€1,580,853,941; $1,764,807,854) and the minimal cost was achieved in 2010, 151,333,139,835 RSD (€1,434,464,541; $1,908,843,843). Conclusion The cost of individual health care in the Republic of Serbia in the period 2010-2015 increased by thirty percent. The highest amount was allocated to treat people with diseases of the circulatory system.

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive relationships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite element formulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include the localised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits further the Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representative volume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, a geometrically nonlinear finite element approximation for highly localized deformation in structures undergoing material failure in the form of strain softening, is developed. The basis for its numerical implementation in this class of problems is defined through the elaboration of Strong Discontinuity Approach-fundamentals. Proposed numerical model uses an Enhanced Assumed Strain Concept for the additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, this part of the deformation mapping is condensed out at the material level, without the application of static condensation technique. The resulting set of constitutive equations is formally identical to that of standard plasticity and therefore, can be solved using the return-mapping algorithm. No assumptions regarding the interface law connecting the displacement discontinuity with the conjugate traction vector are made. As a result, the proposed numerical solution can be applied to a broad range of different mechanical problems including mode-I fracture in brittle materials or the analysis of shear bands. Kinematics induced by strong discontinuitiesIn what follows, a body Ω ⊂ R 3 is separated by a hyperplane ∂ s Ω ⊂ Ω of class C 1 (piecewise). Based on this partition, an approximation of the displacement field of the typeis adopted. In Equation (1), H s denotes the indicating function of the subset Ω + , [ [u] ] the displacement discontinuity and ϕ a smooth ramp function (see [1]). By applying the generalized derivative D(•) to the deformation mapping defined by Equation (1), the deformation gradient is assumed to be of the typewith δ s representing the DIRAC-delta distribution and N the normal vector of ∂ s Ω. Since the enhanced strains are modeled in an incompatible fashion, it is admissible to neglect the gradient of the displacement jump (GRAD [[u] ] = 0), cf. [1,2]. In the proposed EAS concept, only the fieldū is approximated globally conforming, i.e. by using standard shape functions. Constitutive equations: Cohesive lawsAccording to Equation (2), the strains in Ω ± are regularly distributed. Hence, classical constitutive equations can be applied. For the sake of simplicity, we assume a hyperelastic material response characterized by the energy functionalHere, C is the right CAUCHY-GREEN tensor and λ, µ are the LAMÉ constants. The inelastic deformations are taken into account by means of a cohesive law of the type T | ∂sΩ =:in terms of the traction vector T and the displacement jump. The coupling between the constitutive equations in Ω ± and those in ∂ s Ω is provided by the condition of traction continuity. With the positive definiteness of a norm || • || and applying a pull back operation, this continuity condition readsSince the identityT = C · S · N holds (C · S are the MANDEL stresses), the con...

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