Fiber-reinforced plastics are made of an isotropic polymer matrix reinforced with inorganic or organic fibers. These composites become more and more important in light-weight structures. The major contribution to the internal dissipation in these fibrous composites is due to the isotropic matrix material. However, internal dissipation in the fibers has to be taken into account if Carbon or Kevlar fibers, respectively, are applied, or by using organic fibers, which are applied in the automotive industry, for example. The presented two-phase dissipation model is based on the multiplicative split of the deformation gradient of the composite in an elastic and viscous deformation gradient, as well as on the multiplicative split of the fiber deformation gradient in an elastic and viscous fiber deformation gradient. The corresponding viscous evolution equations are derived by a principle of virtual power together with further mechanical and thermal time evolution equations of a dynamic continuous problem. For instance, in the internal power, we consider free energy functions depending on matrix and fiber invariants. In the external power, the non-negative internal dissipation with respect to a positive-definite matrix viscosity tensor and a positive fiber viscosity parameter, respectively, are introduced. The internal power also includes mixed fields for the thermo-elastic matrix and fiber behaviour. In this way, we obtain a mixed stress- strain formulation, which also avoids locking behavior due to the stiff fibers embedded in the incompressible matrix material. Algorithmic terms in the virtual external power provide the energy-momentum scheme of this two-phase dissipation model. We demonstrate the independent dissipative material behaviour due to the matrix and the fibers in dynamic numerical examples with different mechanical and thermal boundary conditions.

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Textile manufacturing techniques as three-dimensional weaving of carbon yarns or the tailored fiber placement based on fixing carbon rovings on a base material are increasingly used for parts in rotordynamical systems. Examples are blades in turbine fans and pump rotors, disc brake rotors and shafts. These metamaterials allow for lower rotational masses while maintaining stiffness and improving heat conduction properties. This leads in rotordynamical systems to higher possible rotational speeds. Thereby, the direction-dependent heat conduction allows for a more efficient cooling in high temperature environments. A continuum formulation of such metamaterials has to take into account the bending stiffness of the yarns or rovings, because carbon yarns and rovings are made of thousends of carbon fibers. A standard Cauchy-continuum with a free energy based on structural tensors are not able to describe this fiber bending stiffness, because fibers are assumed as infinitely thin strings. This results in a three-point bending test of metamaterials simulated by finite elements to wrong curvature effects, in comparison to experimental data. The bending stiffness of the yarns or rovings can be taken into account by tensor invariants based on the second gradient of the deformation mapping, which leads to a generalized continuum formulation. In this paper, we consider a mixed formulation of a generalized continuum, in which the deformation gradient and the first Piola-Kirchhoff stress tensor as well as the continuum rotation vector, the angular velocity vector and the rotational momentum vector are introduced as independent fields. The corresponding weak forms arise from a principle of virtual power, which allows for energy-momentum time integrations of the considered rotordynamical systems. In this principle, the kinetic energy includes the translational energy of a standard Cauchy continuum based on the linear momentum, and the additional rotational energy with a micro-inertia tensor of a generalized continuum. This leads to two additional balance laws in comparison to a standard Cauchy continuum: the balance law of rotational momentum and the balance law of rotational energy. The exact preservation of these two balance laws along with the preservation of each balance law of a standard Cauchy continuum allows the derived energy-momentum scheme. We demonstrate the behavior of this physically-consistent simulation method by using dynamic simulations of a turbine rotor, a disc brake rotor and a rotating shaft subject to thermal and mechanical loads.

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Fiber-reinforced plastics (FRP) are composite materials made of an isotropic polymer matrix reinforced with organic or inorganic fibers. These materials become more and more important in order to provide light-weight structures. The major contribution to the internal dissipation in FRP is due to the isotropic matrix material. However, internal dissipation in the fibers has to be taken into account if Carbon or Kevlar fibers, respectively, are applied. Therefore, we introduce in this presentation a new two-phase dissipation model at finite strains with viscoelastic behaviour in the matrix and the fibers. This model is based on the multiplicative split of the deformation gradient of the composite in an elastic and viscous deformation gradient, respectively, as well as on the multiplicative split of the fiber deformation gradient in an elastic and viscous fiber deformation gradient, respectively. However, the time-dependent matrix behaviour depends directly on the principal invariants of the symmetric elastic right Cauchy-Green tensor. Analogously, the trace of the elastic fiber right Cauchy-Green tensor determines the time evolution of the viscous fiber deformation. We consider free energy functions, depending on these matrix and fiber invariants, in the virtual internal power of a mixed principle of virtual power. In the virtual external power, the non-negative internal dissipation with respect to a positive-definite viscosity tensor or a positive fiber viscosity parameter, respectively, is introduced. In this way, we derive the viscous evolution equations by a variation. The virtual internal power also includes mixed fields for the thermo-elastic matrix and fiber behaviour. Algorithmic terms in the virtual external power lead to an energy-momentum scheme, which provide energy-momentum time integrations of this two-phase model.

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Three-dimensional woven carbon-fiber reinforced polymers (3D-CFRP) are being increasingly used for blades in rotordynamical systems as turbine fans and wind turbines as well as disc brake rotors. These light-weight materials allow for lower rotational masses while maintaining stiffness and improving heat conduction properties. This leads in turbine fans to higher possible rotational speeds, and allows for longer and thus more efficient blades in wind turbines. Thereby, the direction-dependent heat conduction allows for a more efficient cooling in high temperature environments. By using dynamic mixed finite-element methods, these materials can be simulated as anisotropic continua with non-isothermal behaviour in the matrix as well as the fiber material. On the other hand, the rotordynamical systems in which these 3D-CFRP are applied are often formulated by variational approaches. Simulation techniques building on these approaches are preferable. Therefore, we present a novel dynamic mixed finite-element method which is variational-based, and therefore capable to simulate these materials in rotordynamical variational formulations. This dynamic mixed finite-element method preserves each basic balance law emanating from the variational principle exact in a discrete setting. This leads to more physically-consistent results concerning the thermo-mechanical coupling effects. By using independent fields for the deformation of the matrix and the fibers, we also avoid unphysical locking effects. We demonstrate this physically-consistent simulation behaviour by using dynamic simulations of a turbine rotor, a disc brake rotor and a rotating heat-pipe subject to thermal and mechanical loads.

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We present noval higher-order accurate energy-momentum schemes for fiber-reinforced thermo-viscoelastic materials, which additionally includes a new locking-free finite element approximation in space. The well-known locking effect of a standard displacement approximation is avoided by approximating the volume as well as the fiber dilatation with independent fields. Both, the energy-momentum time integration as well as the locking-free finite element approximation are derived by a multi-field principle of virtual power. This variational principle acts as a `smart interface' between the energy-momentum consistent time approximation and the mixed finite element approximation in space. Mechanical as well as thermal fields are approximated independently by higher-order finite elements in space as well as in time. The considered material formulation takes into account thermal volume expansion and heat conduction of a visco-elastic matrix material, and an independent thermal expansion as well as heat conduction of unidirectional fibers. In comparison to a standard time integration with standard finite elements, the fields of deformation, linear momentum, temperature, entropy density as well as volume and fiber strains and stresses are approximated independently in space and time. As numerical examples, we show motions of thin-walled thermo-mechanical structures.

Presentation Paper

In this paper, we present a new higher-order accurate energy-momentum time integration of fiber-reinforced thermo-viscoelastic continua. The energy-momentum schemes are derived from a multi-field principle of virtual power, in which mechanical as well as thermal fields are approximated independently by higher-order finite elements in space as well as in time. In comparison to Gross M. and Dietzsch J. (2017) Variational-based energy-momentum schemes of higher-order for elastic fiber-reinforced continua, Comput. Methods Appl. Mech. Engrg., 320: 509--542, we also consider thermal volume expansion of a visco-elastic matrix material and thermal expansion of unidirectional fibers, as well as tranversely-isotropic heat conduction due to Al-Kinani R. (2014) Thermo-mechanical coupling of transversely isotropic materials using high-order finite elements, Phd-thesis, Faculty of Mathematics/Computer Science and Mechanical Engineering, Clausthal University of Technology. Therefore, in addition to the linear momentum and the deformation, the entropy and the temperature are approximated independently in space and time. The used multi-field principle of virtual power also introduces the strains and stresses of the matrix as well as the fiber strains and stresses as independent variables. As we consider an isochoric-volumetric decoupling of the free energy function of the matrix material, the volumetric strain and stress fields are approximated independently by spatial and temporal finite elements. In this way, the third and fourth tensor invariant of the right-Cauchy Green tensor and the structural tensor of the considered tranversely-isotropic material are independent fields and discretized independently in space and time together with their associated dual variables. We show numerical examples with transient Dirichlet and Neumann boundary conditions.

Presentation Paper

In this presentation, novel higher-order accurate energy-momentum schemes are presented, emanating from a discrete mixed principle of virtual power. These time-stepping schemes are designed for simulating an uni-directional fiber-reinforced material, considered as transversally isotropic nonlinear continua. The matrix material is considered as an isotropic thermo-viscoelastic material and the fibers behave thermo- elastic. Hence, the model takes into account an independent conduction of heat according to Duhamel's law with a transversally isotropic conductivity tensor as well as an independent heat expansion and heat capacity of the matrix and the fibers. The higher-order accurate energy-momentum schemes preserve each balance law of the continuous problem also in the discrete setting, independent of the chosen time step size and the prescribed Neumann and Dirichlet boundary conditions. Therefore, the implemented time step size control with the iteration number as target function does not influence the preservation properties of the time-stepping schemes. The balance laws are also preserved together with different time scales in the mechanical, thermal and viscous time evolution. By calculating each generalized reaction on the boundary, numerical examples verify the energy-momentum consistency of the schemes applied to dynamic simulations of different transient Dirichlet and Neumann boundary conditions.

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The most widely-used materials in mechanical engineering are metals, which are able to dissipate relatively well generated heat as well as to conduct arising strong forces. But, solids made of metals are usally not used in leight-weight structures due to their considerable weight. In this case, metal foams or fiber-reinforced polymers are the materials of choice. In this paper, we consider polymers reinforced by one familie of fibers conducting heat as well as strong forces independently in two directions. The heat conduction model is based on Duhamel's law of transversely isotropic heat conduction, which prescribe heat conduction in fiber direction with an own conductivity coefficient as well as heat conduction normal to the fibers in the matrix material. The heat capacity coefficient of the fiber-reinforced material is assumed linear in temperature. The thermal expansion of the matrix material is defined as volume change depending linearly on the temperature, but the independent thermal expansion of the fibers are assumed as length change depending linearly on the temperature. We simulate nonlinear rotordynamics examples as fast rotating bodies with large deformations. Therefore, we model a flexible body as transversely isotropic thermoelastic continuum, and discredize it by linear or quadratic finite elements. The transient simulation is performed by a higher-order accurate time integration method, called ehG method, which determines independently of the time step size a non-positive thermal dissipation arising from Duhamel's law, and therefore fulfills the corresponding stability estimate based on the Lyapunov function of thermoelasticity numerically exactly.

Presentation Paper

In this paper, a new class of time-stepping schemes for structural dynamics is presented, which originally emanate from Gauss-Runge-Kutta schemes as traditional representatives of higher-order symplectic-momentum schemes. The presented time stepping schemes belong to the family of higher-order energy-momentum schemes, which represent Gauss-Runge-Kutta schemes with a physically motivated time approximation of the considered mechanical system. As higher-order energy-momentum schemes so far are not derived by using a straight-forward design method, a variational-based design of energy-momentum schemes is shown. Here, a differential variational principle of continuum mechanics, Jourdain's principle, is discretized, and energy-momentum schemes emanate as discrete Euler-Lagrange equations. This procedure is strong related, but is not identical, to the derivation of variational integrators (VI), which emanate from discretising a Lagrange function or Hamilton's principle, respectively. Furthermore, this design procedure is well suited to connect energy-momentum schemes with numerical modifications based on mixed variational principles, as the enhanced assumed strain elements for improving the spatial discretisation in direction of a locking-free discrete formulation. Therefore, a Q1/E9 energy-momentum scheme of higher order for the continuum formulation of fiber-reinforced materials is presented. This material formulation is important for simulating dynamics of light-weight structures.

Presentation Paper

For many years, the importance of fiber-reinforced polymers is steadily increasing in mechanical engineering. According to the high strength in fiber direction, these composites replace more and more traditional homogeneous materials, especially in lightweight structures. Fiber-reinforced material parts are often manufactured from carbon fibers as pure attachment parts, or from steel for transmitting forces. Whereas attachment parts are mostly subjected to small deformations, force transmission parts usually suffer large deformations in at least one direction. For the latter, a geometrically non-linear formulation of these anisotropic continua is indispensable. A familar example is a rotor blade, in which the fibers possess the function of stabilizing the structure in order to counteract large centrifugal forces. For long-run numerical analyses of rotor blade motions, we have to apply numerically stable and robust time integration schemes for anisotropic continua. This paper is an extension of Erler N and Gross M (2015) Energy-momentum conserving higher-order time integration of nonlinear dynamics of finite elastic fiber-reinforced continua. Computational Mechanics 55(5):921--942, which is in turn an extension of Gross M and Betsch P and Steinmann P (2005) Conservation properties of a time FE method. Part IV: higher order energy and momentum conserving schemes. Int J Numer Methods Engng 63:1849--1897 to a special anisotropic material class, namely a transversely isotropic hyperelastic material based on the wellknown concept of structural tensors. In the latter Reference, higher-order accurate time-stepping schemes are developed systematically with the focus on numerical stability and robustness in the presence of stiffness combined with large rotations for computing large motions. In the former work, these advantages over conventional time stepping schemes are combined with highly non-linear anisotropic material formulated with polyconvex free energy density functions. The corresponding time integrators preserve all conservation laws of a free motion of a hyperelastic continuum, which means the total linear and the total angular momentum conservation law as well as the total energy conservation law. Both are numerically advantageous, because it guarantees that the discrete configuration vector is embedded in the physically consistent solution space. In order to guarantee the preservation of the total energy, the transient approximation of the anisotropic stress tensor is superimposed with an algorithmic stress field based on an assumed 'strain' field. The presented numerical examples show the behaviour of the non-linear anisotropic material in Schröder J and Neff P and Balzani D (2005) A variational approach for materially stable anisotropic hyperelasticity. Int J Sol Struc 42:4352--4371 under static and transient loads, their conservation laws and the higher-order accuracy of the variational based time approximation.

Presentation Paper

For many years, the application of fiber-reinforced materials is steadily increasing in mechan- ical engineering. These composites replace more and more traditional homogeneous materials in lightweight constructions. Fiber-reinforced material parts are often manufactured from car- bon fibers as pure attachment parts, or from steel for transmitting forces. Whereas attachment parts are mostly subjected to small deformations, force transmission parts usually suffer large deformations. For the latter, a geometrically non-linear formulation of these anisotropic con- tinua is indispensable. A well-known example is a rotor blade, in which the fibers possess the function of stabilizing the structure. For long-run numerical analyses of rotor blade motions, we have to apply numerically stable and robust time integration schemes for anisotropic con- tinua. This work is an extension of the publication Erler N, Groß M (2015) Energy-momentum con- serving higher-order time integration of non-linear dynamics of finite elastic fiber-reinforced continua, Computational Mechanics, Volume 55, Issue 5, pp 921–942, which is in turn an extension of the publication Groß M, Betsch P, Steinmann P (2005) Conservation properties of a time FE method. Part IV: higher order energy and momentum conserving schemes. Int J Numer Methods Eng 63:1849–1897 to a special anisotropic material class, namely a transver- sally isotropic hyper-elastic material. In the latter work, higher-order accurate time integration schemes are developed, with the focus on numerical stability and robustness in the presence of stiffness combined with large rotations. In the former work, these advantages over standard time-stepping schemes are combined with highly non-linear anisotropic material behaviour. The corresponding time integrators preserve all conservation laws of a free motion of a hyper- elastic continuum, which means the total linear and total angular momentum as well as the total energy conservation law. Both are numerically advantageous, because it guarantees that the discrete configuration vector is embedded in the physically consistent solution space. In order to guarantee the preservation of the total energy, the transient approximation of the anisotropic stress tensor is enhanced in connection with assumed strains.

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This paper deals with an energy-entropy-consistent time integration of a thermo-viscoelastic continuum in Poissonian variables. The four differential evolution equations of first order are transformed by a new GENERIC format into a matrix-vector notation. Since the entropy is a primary variable we include thermal constraints to affect the temperatures at the boundaries. This enhanced GENERIC format with thermal constraints yields with the related degeneracy conditions structure preservation properties for a system with thermal constraints. The properties of an isolated system are in addition to a constant linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermo-viscoelastic systems and also for unisolated systems without loads valid. The discretization in time is done with a new TC (Thermodynamically Consistent) integrator. This enhanced TC integrator is constructed such, that the algorithmic structural properties after the discretization in time reflect the underlying enhanced GENERIC format with thermal constraints. As discretization in space the Finite-Element-Method is used. A projection of the test function of the thermal evolution equation is necessary for an energy-consistent discretization. The enhanced GENERIC format with thermal constraints, which is here given in the strong evolution equations, contains the external loads. This yields the necessary weak evolution equations for the solution of the system. The consistency properties are shown for representative numerical examples with different boundary conditions. The coupled mechanical system is solved with a multi-level Newton-Raphson method, based on a consistent linearization.

Presentation Paper

This paper presents an energy-entropy-consistent time stepping scheme for finite thermo-viscoelasticity in Poissonian variables. The four time evolution equations of first order are transformed by a new General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) format into a matrix-vector format. As the entropy is a primary variable, we include algebraic constraints to affect the temperatures at the boundaries. This enhanced GENERIC format with thermal constraints yields with the related degeneracy conditions structure preservation properties for a closed and a open thermomechanical system. The properties of a closed system are in addition to a constant linear and angular momentum, the constant total energy, an increasing total entropy and a decreasing Lyapunov function. The last one is a stability criterion for thermomechanical systems and also for open systems without loads valid. The discretization in time is done with a new TC (Thermodynamically Consistent) integrator. This enhanced TC integrator is constructed such, that the algorithmic structural prop- erties after the discretization in time reflect the underlying enhanced GENERIC format with thermal constraints. The discretization in space is performed by the Finite-Element-Method. A projection of the spatial test function of the thermal evolution equation is necessary for an energy-consistent discretization. The enhanced GENERIC format with thermal constraints, which is here given in the strong evolution equations, contains the external loads. This yields the necessary weak evolution equations for the solution of the system. The consistency prop- erties are shown for representative numerical examples with different boundary conditions. The nonlinear algebraic equations of this thermomechanically coupled system is solved with a multi-level Newton-Raphson method based on a consistent linearization.

Presentation Paper

Presently, there is a significant effort to find more robust and reliable time integration methods for thermo-mechanically coupled dynamics. One possible way of designing such methods is the structure-preserving time integration. The goal here is a method, which ensures that physical features of the underlying problem are also inherited by the discrete solution. Structure-preserving methods have already proven their excellent robustness as well as their reliability of finding meaningful solutions of different thermo-mechanically coupled problems. However, the ability of these methods to take into account different time scales has hitherto based on a fractional step method. Using such a method, we obtain a so-called h-adaption of the time axis, because the problem is solved sequentially on so-called micro time steps. An alternative way is to use a Galerkin-based structure-preserving method in order to increase the degree of the shape functions in time, i.e. a p-adaption in time. Here the problem is solved simultaneously on so-called macro time steps. In this paper, we present such a finite element method in time for thermo-viscoelastodynamics.

Presentation Paper

Presently, there is a significant effort to find more robust and reliable time integration methods for thermo-mechanically coupled dynamics. One possible way of designing such methods is the structure-preserving time integration. The goal here is to design a method, which ensures that physical features of the underlying problem are inherited by the discrete solution. Structure-preserving methods have already proven their excellent robustness as well as their reliability of finding meaningful solutions of different thermo-mechanical problems. However, the ability of these methods to take into account different time scales in the different sub-solutions has hitherto based on a fractional step method. Using a fractional step method, we obtain a so-called h-adaption of the time axis, because the problem is solved sequentially on so-called micro time steps. An alternative way is to use a Galerkin-based structure-preserving method in order to increase the degree of the ansatz functions in time. This corresponds to a p-adaption, which solves the problem simultaneously on all micro time steps. In this paper, we present a Galerkin-based method for problems with different time scales such as finite thermo-viscoelastodynamics. Here, the thermal and the mechanical time scales, respectively, are usually different, where the mechanical time scales are the motion and the internal variables.

Presentation Paper

Heutzutage ist bei der Entwicklung technischer Produkte die Optimierung der verwendeten Werkstoffe eine wichtige Aufgabe. Deshalb kommen immer öfter Materialien zum Einsatz deren physikalische Eigenschaften, wie Leitfähigkeit und Deformationsverhalten, weitgehend einstellbar sind. Zu diesen Materialien zählen die sogenannten Kunststoffe oder Polymere, die man gewöhnlich in Elastomere, Duroplaste und Thermoplaste einteilt. Neuere Entwicklungen im Automobilbereich sind zum Beispiel mikrozellulare Elastomere als Fahrwerksfedern oder Duroplastschäume als Dämmmatten im Motorraum. Im Elektronikbereich werden neuerdings leitfähige Thermoplaste für Kühlkörper und Wärmerohre eingesetzt. Weiterhin wird bei der Produktentwicklung immer mehr auf den Kosten- und Zeitfaktor geachtet. Neue Produkte sollen schneller und kostengünstiger entwickeln werden. Dazu werden virtuelle Versuche in Form von Finite-Elemente-Simulationen betrachtet, und experimentelle Versuche zur Validierung der numerischen Resultate herangezogen. Ein typisches Beispiel ist die Fahrwerksentwicklung im Automobilbereich. Somit entsteht eine kombinierte Anwendung numerischer und experimenteller Verfahren. Damit numerische Simulationen realistische Resultate liefern können, müssen neben den zugrundeliegenden diskreten Gleichungen auch die zu deren Lösung verwendeten numerischen Verfahren physikalisch konsistent sein. Diese Konsistenz sollte unabhängig von Diskretisierungsparametern wie Elementgrösse und Zeitschrittweite sein, damit die Simulationszeit reduziert werden kann. Somit entsteht der Wunsch sowohl nach einer versteifungsfreien räumlichen Diskretisierung, als auch nach einer energie-impuls-konsistenten zeitlichen Diskretisierung. Diese spezielle Zeitdiskretisierung garantiert selbst bei grossen Zeitschrittweiten die geforderte physikalische Konsistenz, als auch die gewünschte numerische Stabilität. Dieser Vortrag befasst sich im Hauptteil mit der energie-impuls-konsistenten Zeitdiskretisierung für mikrozellularer Elastomere. Die zu bewahrenden physikalischen Eigenschaften folgen aus der Modellierung des nichtlinearen viskoelastischen Verhaltens, des thermo-viskoelastischen Aufheizens, sowie aus einer isotropen Wärmeleitung und -speicherung.

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