Homogenization of Fatigue and Creep Durability of Composites The paper is an extension of the the... more Homogenization of Fatigue and Creep Durability of Composites The paper is an extension of the theory presented in [4] to fatigue, creep and other types of time-dependent material behaviour. In the previous paper a multi-phase periodic composite subjected to prescribed mechanical loads was considered. The asymptotic homogenization was applied for calculation of homogenized macro-stresses. A nonlocal approximate macro-strength condition, defined on homogenised stress-field, was derived from the micro-strength conditions and their convergence to the approximate macro-strength condition, as the structure period tends to zero, was proved. This paper presents the analogous results for the macro-durability of elastic and viscoelastic composites.
We consider the elasticity problem in a heterogeneous domain with \(\varepsilon -\) periodic micr... more We consider the elasticity problem in a heterogeneous domain with \(\varepsilon -\) periodic micro-structure, \(\varepsilon \ll 1\), including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, but are locked to a matrix on a piece of the boundary. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005). A time discretization scheme from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005) reduces the contact problem to the frictional traction known from the previous step on each time-increment and is then solved by fixed point argument. For a fixed \(\varepsilon\), the necessary condition for the keeping the contact continuous in time (for the contraction mapping) is given in (Cocu and Rocca, Existence results for unilateral quasistaric contact problems with fricrtion and adhesion, 2000) and (Eck et al., Unilateral contact problems variational methods and existence theorems. Springer, 2005) in the form of the bound on the frictional coefficient by lower and upper bounds on the elastic tensor and norms of the direct and inverse trace operators. We further look for the spatial homogenization of the contact problems on each time-increment and introduce scaling of Sobolev–Slobodetsy norms and Bessel potentials. By shifting argument we obtain the preliminary estimates for normal tractions in a better space and proof its strong convergence. The limiting energy and the dissipation term in the stability condition obtained for the contact with Tresca’s friction law in (Cioranescu et al., Asymptotic Anal. 82(3–4), 2013) are then valid also for the Coulomb one. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005), for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time.
Progress in Industrial Mathematics at ECMI 2000, 2002
A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads includ... more A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads including prescribed interface jumps of displacements and tractions is considered. The composite components are anisotropic linear viscoelastic and aging (described by the non-convolution Volterra integral operators). The paper presents some results about solvability and uniqueness of solution to this problem in the class of continuous Banach-valued functions with values in the Sobolev spaces, asymptotic homogenisation, and uniform in time 2-scale convergence in appropriate functional spaces. and the part of boundary aDn, has zero measure on each an,. Let i,j, h, k = 1, 2, ... , n, and summation from 1 ton over the repeating subscripts is assumed hereafter. For a solid n, for any t E [0, T], we consider the following problem: a~h ([Qihjk(x/u~~:• •)] (t) + a~h(x, t)) = fio(x, t), x E n \ E (1) Ui(X, t) = Xi(X, t), X E aDn, (2) ([Qihjk(x) au~~:'•)] (t) + O'~h(x, t))nh(x) = Wi(x, t), X E aNn, (3)
The theory of the two-scale convergence was applied to homogenization of elasto-plastic composite... more The theory of the two-scale convergence was applied to homogenization of elasto-plastic composites with a periodic structure and exponential hardening law. The theory is based on the fact that the elastic as well as the plastic part of the stress field two-scale converges to a limit, which is factorized by parts, depending only on macroscopic characteristics, represented in terms of corresponding part of the homogenised stress tensor and only on stress concentration tensor, related to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to metallic matrix material with Ludwik and Hocket-Sherby hardening law and pure elastic inclusions in two numerical examples. Results were compared with results of mechanical averaging based on the self-consistent methods.
Mathematical Methods in the Applied Sciences, 2019
We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The... more We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For thi...
ABSTRACT We consider a linear elasticity boundary value problem in a beam with Robin boundary con... more ABSTRACT We consider a linear elasticity boundary value problem in a beam with Robin boundary condition at an end and on a segment of the lateral boundary in the middle of the beam. The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard asymptotic approach with an additional expansion suggested to fulfil the Robin conditions. The 3D Robin conditions result into 1D Robin boundary conditions for corresponding ODEs. The asymptotic error is estimated and illustrated by a numerical comparison of the 3D and 1D solutions.
Journal of the Mechanical Behavior of Materials, 2009
An asymptotic homogenization procedure is employed to obtain effective constitutive law, a homoge... more An asymptotic homogenization procedure is employed to obtain effective constitutive law, a homogenized macro-stresses and-deformation gradient and a first approximation to the micro-stresses and deformation gradient in non-linear elastic composite materials, from their microstructure, properties of the components and applied macro-loads. Using the approximate micro-deformation gradient, a non-local macrostrength condition for the composite materials is determined by elastic and strength micro-properties of the periodicity cell and expressed in terms of the homogenized macro-deformation gradient. The procedure is illustrated by formal asymptotic expansion for incompressible Neo-Hooke material.
In Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), the dimension of a 3D linea... more In Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), the dimension of a 3D linear elasticity boundary value problem with Robin boundary condition is asymptotically reduced. Assumption 1.4 in Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), leads to a 1D system in which the bending and tensile components are decoupled. With a generalization in this contribution, we obtain a coupled 1D system. We prove that the asymptotic error estimate in Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481) remains true and illustrate the influence of the tension and torsion on the bending by a numerical example.
Journal of the Mechanical Behavior of Materials, 2009
We consider the contact of two elastic bodies with rough surfaces at the interface. The size of t... more We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and-valleys is very small compared with the macrosize of the bodies' domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves an asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domaines. The macrocontact condition is then used in a FEM model for the contact problem at the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition. The approach is illustrated by the application to the hip-prosthesis-bone contact. 2. INTRODUCTION The surface of the cementless implant is coated in such a way that the bone is in direct contact with this surface with the idea that the bone grows into the micro-troughs and pores of the coating to provide enhanced stability and rapid osteointegration. The study of the bone-stem contact by direct application of the finite element method taking into account microgeometry of the interface is very expensive. The theoretical modelling of the bone-implant interface aims to simplify the problem by replacing the interface layer by a smooth contact surface with some effective macro-contact conditions at the expense of a slight decrease in the accuracy of the results predicted. 143
A multi-phase composite with periodically distributed smooth inclusions is considered. The compos... more A multi-phase composite with periodically distributed smooth inclusions is considered. The composite components are linear viscoelastic and aging (of the non-convolution integral Volterra type) and are subjected to isotropic shrinkage. The shrinkage deformation can be associated with a fictitious thermoelastic deformation in the constitutive law. An averaging procedure is presented enabling to obtain effective (homogenized) viscoelastic and shrinkage (thermoelastic) properties of the composite and the homogenized stress - field from known properties of the components. Known homogenization results given by [1], [2] have dealt with viscoelasticity of a rather particular differential form.
ABSTRACT The theory of the two-scale convergence was applied to homogenization of initial flow st... more ABSTRACT The theory of the two-scale convergence was applied to homogenization of initial flow stresses and hardening con-stants in some exponential hardening laws for elasto-plastic composites with a periodic microstructure. The theory is based on the fact that both the elastic and the plastic part of the stress field two-scale converge to a limit, which can be factorized by parts, one of which depends only on the macroscopic, and the other one -only on the microscopic char-acteristics. The first factor is represented in terms of the homogenized stress tensor and the second factor -in terms of stress concentration tensor, that relates to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to a composite, that consists of the metallic elasto-plastic matrix with Ludwik and Hocket-Sherby hardening law and pure elastic silica inclusions. Results were compared with those of averaging based on the self-consistent methods.
We consider the contact of two elastic bodies with rough surfaces at the interface. The size of t... more We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and valleys is very small compared with the macrosize of the bodies’ domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves the twoscale asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domains. The macrocontact condition can then be used in a FEM model for the contact problem on the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition.
Homogenization in Integral Viscoelasticity A multi-phase periodic composite subjected to inhomoge... more Homogenization in Integral Viscoelasticity A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads including prescribed interface jumps of displacements and tractions is considered. The composite components are anisotropic linear viscoelastic and aging (described by the non-convolution Volterra integral operators). The paper presents some results about asymptotic homogenization and 2-scale convergence in appropriate function spaces.
An asymptotic homogenization procedure is employed to obtain effective elastic properties of the ... more An asymptotic homogenization procedure is employed to obtain effective elastic properties of the composite tooth filling, a homogenized macro-stress field and a first approximation to the micro-stress field, from properties of the components and applied macro-loads. Using the approximate micro-stress field, a non-local initial strength and fatigue durability macro-conditions for the composite filling material is expressed in terms of the homogenized macro-stresses. An illustrative example with the stress singularity on the tooth-filling interface is presented showing the need in the non-local analysis. Effective elastic properties of the tooth filling is numerically simulated for some size distributions of the reinforcing particles. 2. INTRODUCTION AND MOTIVATION Tooth filling materials are considered consisting of a polymer matrix filled with glass particles. The volume fraction, size, shape and distribution of the particles influence mechanical properties of the filling material. Since 70-es, the homogenization techniques were widely used for obtaining effective elastic properties of the composite and the homogenized macro-stress field. The simplest averaging theories are mixture theory and Representative Volume Element (RVE) method (Hashin, 1983). In [1], the approximation to the micro-stress-field is derived from properties of the components, micro-geometry of the composite and applied macro-loads, using a
Prior to surgery, correction of the defects was simulated and the results of the virtual procedur... more Prior to surgery, correction of the defects was simulated and the results of the virtual procedure superimposed on a three-dimensional head model of the patients. Tissue elasticity and thickness of the flap, however, could not be ascertained in advance, indicating the limitations of the method. However, three-dimensional visualizing of body parts leads to a more profound understanding of the anatomical structures that need to be reconstructed and hence to an improved quality of surgery planning and final outcome. In our experience, the computer aided surgery can already deliver valuable information facilitating the reconstruction of complex structure defects of the face. However, it is not yet able to replace the judgement of an experienced surgeon. 6056 We-Th, no. 16 (P62) Geometrical changes of the pelvis after Salter osteotomy
The aim of this contribution is to compute the effective in-plane tension and shear behaviour of ... more The aim of this contribution is to compute the effective in-plane tension and shear behaviour of textile-like elastic materials, that is, plates or shells with a periodic micro-structure composed of long woven or knitted fibres. The knitting or weaving results in multiple periodic contact between fibres. Mathematically the problem can be formulated as an in-plane elasticity problem defined in a heterogeneous domain with ε-periodic micro-structure, including multiple microcontact between the structural components, which is described by the Signorini and Tresca-friction contact conditions. The asymptotic analysis and homogenized limit for such problems was recently obtained by Damlamian et al. via periodic unfolding strategy. These results are briefly recalled in the paper. The obtained two-scale algorithm is implemented for some specific textile materials, which are macroscopic shells with a 3-D microstructure including contact. For each macroscopic deformation state, a contact problem in the periodicity cell or representative volume element is solved and the corresponding non-linear macroscopic stress-strain relation is obtained. The results are illustrated by the simulation of woven and knitted textiles.
Homogenization of Fatigue and Creep Durability of Composites The paper is an extension of the the... more Homogenization of Fatigue and Creep Durability of Composites The paper is an extension of the theory presented in [4] to fatigue, creep and other types of time-dependent material behaviour. In the previous paper a multi-phase periodic composite subjected to prescribed mechanical loads was considered. The asymptotic homogenization was applied for calculation of homogenized macro-stresses. A nonlocal approximate macro-strength condition, defined on homogenised stress-field, was derived from the micro-strength conditions and their convergence to the approximate macro-strength condition, as the structure period tends to zero, was proved. This paper presents the analogous results for the macro-durability of elastic and viscoelastic composites.
We consider the elasticity problem in a heterogeneous domain with \(\varepsilon -\) periodic micr... more We consider the elasticity problem in a heterogeneous domain with \(\varepsilon -\) periodic micro-structure, \(\varepsilon \ll 1\), including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, but are locked to a matrix on a piece of the boundary. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005). A time discretization scheme from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005) reduces the contact problem to the frictional traction known from the previous step on each time-increment and is then solved by fixed point argument. For a fixed \(\varepsilon\), the necessary condition for the keeping the contact continuous in time (for the contraction mapping) is given in (Cocu and Rocca, Existence results for unilateral quasistaric contact problems with fricrtion and adhesion, 2000) and (Eck et al., Unilateral contact problems variational methods and existence theorems. Springer, 2005) in the form of the bound on the frictional coefficient by lower and upper bounds on the elastic tensor and norms of the direct and inverse trace operators. We further look for the spatial homogenization of the contact problems on each time-increment and introduce scaling of Sobolev–Slobodetsy norms and Bessel potentials. By shifting argument we obtain the preliminary estimates for normal tractions in a better space and proof its strong convergence. The limiting energy and the dissipation term in the stability condition obtained for the contact with Tresca’s friction law in (Cioranescu et al., Asymptotic Anal. 82(3–4), 2013) are then valid also for the Coulomb one. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from (Mielke and Rossi, Existence and uniqueness results for general rate-independent hysteresis problems, 2005), for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time.
Progress in Industrial Mathematics at ECMI 2000, 2002
A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads includ... more A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads including prescribed interface jumps of displacements and tractions is considered. The composite components are anisotropic linear viscoelastic and aging (described by the non-convolution Volterra integral operators). The paper presents some results about solvability and uniqueness of solution to this problem in the class of continuous Banach-valued functions with values in the Sobolev spaces, asymptotic homogenisation, and uniform in time 2-scale convergence in appropriate functional spaces. and the part of boundary aDn, has zero measure on each an,. Let i,j, h, k = 1, 2, ... , n, and summation from 1 ton over the repeating subscripts is assumed hereafter. For a solid n, for any t E [0, T], we consider the following problem: a~h ([Qihjk(x/u~~:• •)] (t) + a~h(x, t)) = fio(x, t), x E n \ E (1) Ui(X, t) = Xi(X, t), X E aDn, (2) ([Qihjk(x) au~~:'•)] (t) + O'~h(x, t))nh(x) = Wi(x, t), X E aNn, (3)
The theory of the two-scale convergence was applied to homogenization of elasto-plastic composite... more The theory of the two-scale convergence was applied to homogenization of elasto-plastic composites with a periodic structure and exponential hardening law. The theory is based on the fact that the elastic as well as the plastic part of the stress field two-scale converges to a limit, which is factorized by parts, depending only on macroscopic characteristics, represented in terms of corresponding part of the homogenised stress tensor and only on stress concentration tensor, related to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to metallic matrix material with Ludwik and Hocket-Sherby hardening law and pure elastic inclusions in two numerical examples. Results were compared with results of mechanical averaging based on the self-consistent methods.
Mathematical Methods in the Applied Sciences, 2019
We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The... more We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For thi...
ABSTRACT We consider a linear elasticity boundary value problem in a beam with Robin boundary con... more ABSTRACT We consider a linear elasticity boundary value problem in a beam with Robin boundary condition at an end and on a segment of the lateral boundary in the middle of the beam. The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard asymptotic approach with an additional expansion suggested to fulfil the Robin conditions. The 3D Robin conditions result into 1D Robin boundary conditions for corresponding ODEs. The asymptotic error is estimated and illustrated by a numerical comparison of the 3D and 1D solutions.
Journal of the Mechanical Behavior of Materials, 2009
An asymptotic homogenization procedure is employed to obtain effective constitutive law, a homoge... more An asymptotic homogenization procedure is employed to obtain effective constitutive law, a homogenized macro-stresses and-deformation gradient and a first approximation to the micro-stresses and deformation gradient in non-linear elastic composite materials, from their microstructure, properties of the components and applied macro-loads. Using the approximate micro-deformation gradient, a non-local macrostrength condition for the composite materials is determined by elastic and strength micro-properties of the periodicity cell and expressed in terms of the homogenized macro-deformation gradient. The procedure is illustrated by formal asymptotic expansion for incompressible Neo-Hooke material.
In Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), the dimension of a 3D linea... more In Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), the dimension of a 3D linear elasticity boundary value problem with Robin boundary condition is asymptotically reduced. Assumption 1.4 in Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481), leads to a 1D system in which the bending and tensile components are decoupled. With a generalization in this contribution, we obtain a coupled 1D system. We prove that the asymptotic error estimate in Bare et al. (Appl. Anal., 2013, doi:10.1080/00036811.2013.823481) remains true and illustrate the influence of the tension and torsion on the bending by a numerical example.
Journal of the Mechanical Behavior of Materials, 2009
We consider the contact of two elastic bodies with rough surfaces at the interface. The size of t... more We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and-valleys is very small compared with the macrosize of the bodies' domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves an asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domaines. The macrocontact condition is then used in a FEM model for the contact problem at the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition. The approach is illustrated by the application to the hip-prosthesis-bone contact. 2. INTRODUCTION The surface of the cementless implant is coated in such a way that the bone is in direct contact with this surface with the idea that the bone grows into the micro-troughs and pores of the coating to provide enhanced stability and rapid osteointegration. The study of the bone-stem contact by direct application of the finite element method taking into account microgeometry of the interface is very expensive. The theoretical modelling of the bone-implant interface aims to simplify the problem by replacing the interface layer by a smooth contact surface with some effective macro-contact conditions at the expense of a slight decrease in the accuracy of the results predicted. 143
A multi-phase composite with periodically distributed smooth inclusions is considered. The compos... more A multi-phase composite with periodically distributed smooth inclusions is considered. The composite components are linear viscoelastic and aging (of the non-convolution integral Volterra type) and are subjected to isotropic shrinkage. The shrinkage deformation can be associated with a fictitious thermoelastic deformation in the constitutive law. An averaging procedure is presented enabling to obtain effective (homogenized) viscoelastic and shrinkage (thermoelastic) properties of the composite and the homogenized stress - field from known properties of the components. Known homogenization results given by [1], [2] have dealt with viscoelasticity of a rather particular differential form.
ABSTRACT The theory of the two-scale convergence was applied to homogenization of initial flow st... more ABSTRACT The theory of the two-scale convergence was applied to homogenization of initial flow stresses and hardening con-stants in some exponential hardening laws for elasto-plastic composites with a periodic microstructure. The theory is based on the fact that both the elastic and the plastic part of the stress field two-scale converge to a limit, which can be factorized by parts, one of which depends only on the macroscopic, and the other one -only on the microscopic char-acteristics. The first factor is represented in terms of the homogenized stress tensor and the second factor -in terms of stress concentration tensor, that relates to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to a composite, that consists of the metallic elasto-plastic matrix with Ludwik and Hocket-Sherby hardening law and pure elastic silica inclusions. Results were compared with those of averaging based on the self-consistent methods.
We consider the contact of two elastic bodies with rough surfaces at the interface. The size of t... more We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and valleys is very small compared with the macrosize of the bodies’ domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves the twoscale asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domains. The macrocontact condition can then be used in a FEM model for the contact problem on the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition.
Homogenization in Integral Viscoelasticity A multi-phase periodic composite subjected to inhomoge... more Homogenization in Integral Viscoelasticity A multi-phase periodic composite subjected to inhomogeneous shrinkage and mechanical loads including prescribed interface jumps of displacements and tractions is considered. The composite components are anisotropic linear viscoelastic and aging (described by the non-convolution Volterra integral operators). The paper presents some results about asymptotic homogenization and 2-scale convergence in appropriate function spaces.
An asymptotic homogenization procedure is employed to obtain effective elastic properties of the ... more An asymptotic homogenization procedure is employed to obtain effective elastic properties of the composite tooth filling, a homogenized macro-stress field and a first approximation to the micro-stress field, from properties of the components and applied macro-loads. Using the approximate micro-stress field, a non-local initial strength and fatigue durability macro-conditions for the composite filling material is expressed in terms of the homogenized macro-stresses. An illustrative example with the stress singularity on the tooth-filling interface is presented showing the need in the non-local analysis. Effective elastic properties of the tooth filling is numerically simulated for some size distributions of the reinforcing particles. 2. INTRODUCTION AND MOTIVATION Tooth filling materials are considered consisting of a polymer matrix filled with glass particles. The volume fraction, size, shape and distribution of the particles influence mechanical properties of the filling material. Since 70-es, the homogenization techniques were widely used for obtaining effective elastic properties of the composite and the homogenized macro-stress field. The simplest averaging theories are mixture theory and Representative Volume Element (RVE) method (Hashin, 1983). In [1], the approximation to the micro-stress-field is derived from properties of the components, micro-geometry of the composite and applied macro-loads, using a
Prior to surgery, correction of the defects was simulated and the results of the virtual procedur... more Prior to surgery, correction of the defects was simulated and the results of the virtual procedure superimposed on a three-dimensional head model of the patients. Tissue elasticity and thickness of the flap, however, could not be ascertained in advance, indicating the limitations of the method. However, three-dimensional visualizing of body parts leads to a more profound understanding of the anatomical structures that need to be reconstructed and hence to an improved quality of surgery planning and final outcome. In our experience, the computer aided surgery can already deliver valuable information facilitating the reconstruction of complex structure defects of the face. However, it is not yet able to replace the judgement of an experienced surgeon. 6056 We-Th, no. 16 (P62) Geometrical changes of the pelvis after Salter osteotomy
The aim of this contribution is to compute the effective in-plane tension and shear behaviour of ... more The aim of this contribution is to compute the effective in-plane tension and shear behaviour of textile-like elastic materials, that is, plates or shells with a periodic micro-structure composed of long woven or knitted fibres. The knitting or weaving results in multiple periodic contact between fibres. Mathematically the problem can be formulated as an in-plane elasticity problem defined in a heterogeneous domain with ε-periodic micro-structure, including multiple microcontact between the structural components, which is described by the Signorini and Tresca-friction contact conditions. The asymptotic analysis and homogenized limit for such problems was recently obtained by Damlamian et al. via periodic unfolding strategy. These results are briefly recalled in the paper. The obtained two-scale algorithm is implemented for some specific textile materials, which are macroscopic shells with a 3-D microstructure including contact. For each macroscopic deformation state, a contact problem in the periodicity cell or representative volume element is solved and the corresponding non-linear macroscopic stress-strain relation is obtained. The results are illustrated by the simulation of woven and knitted textiles.
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Papers by J. Orlik