Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2017"

Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua”** by E. Hellinger

Abstract: The fundamental review article DIE ALLGEMEINEN ANSATZE DER MECHANIK DER KONTINUA in the Encyklopadie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV-4, Hft. 5 (1913) by Ernst Hellinger has not been translated into English so far. We believe that such a circumstance is really deprecative, as the insight reached by Hellinger in the mathematical structure of continuum mechanics seems, in some aspects, unsurpassed even nowadays. Hellinger's scientific manuscripts do not fill more than one and half boxes in library storage [2], but their impact on mathematics and mechanical sciences is profound. Indeed, the Hellinger-Reissner variational principle is still a fundamental tool in theoretical and numerical mechanics. The intent of this paper is threefold: i) to allow to those who cannot understand German to enjoy the reading of a crystal-clear and still topical article whose content has some enlightening parts, ii) to show that only one century ago the principle of virtual work (or virtual velocities) was regarded as the central principle in continuum mechanics and that Hellinger did forecast already then the main lines of its development, iii) to discuss some technical and conceptual aspects of the variational principles in continuum mechanics which some authors consider still controversial.

The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless

Abstract: We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of angular momentum are formulated at an infinitesimal cube considering the total stress as linear and quadratic approximation of a spatial Taylor series expansion.

Peridynamic integrals for strain invariants of homogeneous deformation

Abstract: This study presents the peridynamic integrals. They enable the derivation of the peridynamic (nonlocal) form of the strain invariants. Therefore, the peridynamic form of the existing classical strain energy density functions can readily be constructed for linearly elastic and hyperelastic isotropic materials without any calibration. A general form of the force density vector is derived based on the strain energy density function that is expressed in terms of the first invariant of the right Cauchy-Green strain tensor and the Jacobian. In the case of linear elastic response for isotropic materials, the peridynamic force density vector is derived based on the classical form of the strain energy density function for three- and two-dimensional analysis. Also, a new form of the strain energy density function leads to a force density vector similar to that of bond-based peridynamics. Numerical results concern the verification of the peridynamic predictions with these force density vectors by considering a rectangular plate under uniform stretch.

Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case

Abstract: This paper is devoted to the modeling of compressible hyperelastic materials whose response functions exhibit uncertainties at some scale of interest. The construction of parametric probabilistic representations for the Ogden class of stored energy functions is specifically considered and formulated within the framework of Information Theory. The overall methodology relies on the principle of maximum entropy, which is invoked under constraints arising from existence theorems and consistency with linearized elasticity. As for the incompressible case discussed elsewhere, the derivation essentially involves the conditioning of some variables on the stochastic bulk and shear moduli, which are shown to be statistically dependent random variables in the present case. The explicit construction of the probability measures is first addressed in the most general setting. Subsequently, particular results for classical Neo-Hookean and Mooney-Rivlin materials are provided. Salient features of the probabilistic representations are finally highlighted through forward Monte-Carlo simulations. In particular, it is seen that the models allow for the reproduction of typical experimental trends, such as a variance increase at large stretches. A stochastic multiscale analysis, where uncertainties on the constitutive law of the matrix phase are taken into account through the proposed approach, is also presented.

Free vibration analysis of nanocomposite sandwich plates reinforced with CNT aggregates

Abstract: In this article, free vibration analysis of simply supported sandwich plates with a flexible core and two functionally graded nanocomposite face sheets resting on two-parameter elastic foundation is carried out using Navier's solution. The nanocomposite face sheets are reinforced by randomly oriented and aggregated carbon nanotube (CNT). The material properties of the nanocomposites plates are graded along the thickness and are estimated though the Eshelby–Mori–Tanaka approach. In the proposed theory, 3-D elasticity theory and first order shear deformation theory are used for core and face sheets, respectively. Also, Hamilton's variational principle is used to derive the equations of motion.

Generalized Timoshenko-Reissner models for beams and plates, strongly heterogeneous in the thickness direction

Abstract: The paper is concerned with a thin beam and a thin plate made of a transversally isotropic linearly elastic material heterogeneous in the thickness direction. The ordinary Kirchhoff–Love and Timoshenko–Reissner models are known to be unacceptable for a strongly heterogeneous material or for a multilayered material with large ratio between the Young moduli of layers. A generalized Timoshenko–Reissner model is proposed for bending and free vibrations of such beams/plates. A multilayered beam/plate is reduced to one-layer one with the equivalent elastic parameters. The range of application of such a model is very wide.

Frictional force between a rotationally symmetric indenter and a viscoelastic half-space

Abstract: The paper presents a numerical study of the tangential contact between a rigid indenter and an elastomer with linear rheology. Occurring friction forces are caused exclusively by internal dissipative losses. Especially the combinations of a conical or paraboloid indenter and the Maxwell or standard body with fixed ratio of moduli are studied. One result is, that by the use of proper chosen dimensionless variables, for each of the combinations, the contact forces depend explicitly on the indentation depth only. Curves describing this dependencies are given as Bezier function fits. Application of these methods to an indenter with arbitrary shape and materials with a discrete relaxation spectrum described by Prony series is possible. Although only stationary solutions are analysed here, the method enables us to examine the transient process as well.

A unified beam theory with strain gradient effect and the von Kármán nonlinearity

Abstract: In this paper, the governing equations and finite element formulations for a microstructure-dependent unified beam theory with the von Karman nonlinearity are developed. The unified beam theory includes the three familiar beam theories (namely, Euler-Bernoulli beam theory, Timoshenko beam theory, and third-order Reddy beam theory) as special cases. The unified beam formulation can be used to facilitate the development of general finite element codes for different beam theories. Nonlocal size-dependent properties are introduced through classical strain gradient theories. The von Karman nonlinearity which accounts for the coupling between extensional and bending responses in beams with moderately large rotations but small strains is included. Equations for each beam theory can be deduced by setting the values of certain parameters. Newton's iterative scheme is used to solve the resulting nonlinear set of finite element equations. The numerical results show that both the strain gradient theory and the von Karman nonlinearity have a stiffening effect, and therefore, reduce the displacements. The influence is more prominent in thin beams when compared to thick beams.

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part II: Non-classical energy-minimizing microrotations in 3D and their computational validation†

Abstract: In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field F≔∇φ:Ω→ GL +(n) and the microrotation field R:Ω→ SO (n), the shear–stretch energy is necessarily of the form Wμ,μc(R;F)≔μ sym (RTF−1)2+μc skew (RTF−1)2. We aim at the derivation of closed form expressions for the minimizers of Wμ,μc(R;F) in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension n=3, as a function of F∈ GL +(3). In a previous contribution (Part I), we have first shown that, for all n≥2, the full range of weights μ>0 and μc≥0 can be reduced to either a classical or a non-classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in SO(2) and proved their global optimality. In the present contribution (Part II), we characterize the non-classical optimal rotations in dimension n=3. After a lift of the minimization problem to the unit quaternions, the Euler–Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates rpolar μ,μc±(F)∈ SO (3) which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of SO(3). Geometrically, our proposed optimal Cosserat rotations rpolar μ,μc±(F) act in the plane of maximal stretch. Our previously obtained explicit formulae for planar optimal Cosserat rotations in SO(2) reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear–stretch energy and criteria for the existence of non-classical optimal rotations.

Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge

Abstract: The mathematical models describing equilibrium of cracked elastic plates with rigid thin stiffeners on the outer boundary are studied. On the crack faces the boundary conditions are specified in the form of inequalities which describe the mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the length of the thin rigid stiffener reinforcing the cracked Kirchhoff-Love plate on the outer edge. The existence is proved of the solution to the optimal control problem. For this problem the cost functional is defined by an arbitrary continuous functional, while the length parameter of the thin rigid stiffener is chosen as a control function.

Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model

Abstract: We consider a nonlinear model for a thermoelastic beam that can enter in contact with obstacles. We first prove the well-posedness of this problem. Next, we propose a discretization by Euler and Crank-Nicolson schemes in time and finite elements in space and perform the a priori analysis of the discrete problem. Some numerical experiments are presented.

A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model

Abstract: We consider the weighted isotropic relaxed micromorphic model and provide an in depth investigation of the characteristic dispersion curves when the constitutive parameters of the model are varied. The weighted relaxed micromorphic model generalizes the classical relaxed micromorphic model previously introduced by the authors, since it features the Cartan-Lie decomposition of the tensors P,t and Curl P in their dev sym , skew and spherical part. It is shown that the split of the tensor P,t in the micro-inertia provides an independent control of the cut-offs of the optic banches. This is crucial for the calibration of the relaxed micromorphic model on real band-gap metamaterials. Even if the physical interest of the introduction of the split of the tensor Curl P is less evident than in the previous case, we discuss in detail which is its effect on the dispersion curves. Finally, we also provide a complete parametric study involving all the constitutive parameters of the introduced model, so giving rise to an exhaustive panorama of dispersion curves for the relaxed micromorphic model.

Stress intensity factor for multiple inclined or curved cracks problem in circular positions in plane elasticity

Abstract: The problems of multiple inclined or curved cracks in circular positions is treated by using the hypersingular integral equation method. The cracks center are placed at the edge of a virtual circle with radius R. The first crack is fixed on the x-axis while the second crack is located on the boundary of a circle with the varying angle, θ. A system of hypersingular integral equations is formulated and solved numerically for the stress intensity factor (SIF). Numerical examples demonstrate the effect of interaction between two cracks in circular positions are given. It is found that, the severity at the second crack tips are significant when the ratio length of the second to the first crack is small and it is placed at a small angle of θ.

A quaternion-based model for optimal control of an airborne wind energy system

Abstract: Airborne wind energy systems are capable of extracting energy from higher wind speeds at higher altitudes. The configuration considered in this paper is based on a tethered kite flown in a pumping orbit. This pumping cycle generates energy by winching out at high tether forces and driving a generator while flying figures-of-eight, or lemniscates, as crosswind pattern. Then, the tether is reeled in while keeping the kite at a neutral position, thus leaving a net amount of generated energy. In order to achieve an economic operation, optimization of pumping cycles is of great interest. In this paper, first the principles of airborne wind energy will be briefly revisited. The first contribution is a singularity-free model for the tethered kite dynamics in quaternion representation, where the model is derived from first principles. The second contribution is an optimal control formulation and numerical results for complete pumping cycles. Based on the developed model, the setup of the optimal control problem (OCP) is described in detail along with its numerical solution based on the direct multiple shooting method in the CasADi optimization environment. Optimization results for a pumping cycle consisting of six lemniscates show that the approach is capable to find an optimal orbit in a few minutes of computation time. For this optimal orbit, the power output is increased by a factor of two compared to a sophisticated initial guess for the considered test scenario.

On the non‐stationary non‐Newtonian flow through a thin porous medium

Abstract: We consider a non-stationary incompressible non-Newtonian flow in a thin porous medium of thickness e which is perforated by periodically distributed solid cylinders of size ae. The viscosity is supposed to obey the power law with flow index 32

Moment‐matching based model reduction for Navier–Stokes type quadratic‐bilinear descriptor systems

Abstract: We discuss a Krylov subspace projection method for model reduction of a special class of quadratic-bilinear descriptor systems. The goal is to extend the two-sided moment-matching method for quadratic-bilinear ODEs to descriptor systems in an efficient and reliable way. Recent results have shown that the direct application of interpolation based model reduction techniques to linear descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we transform the quadratic-bilinear descriptor system into an equivalent quadratic-bilinear ODE system for which the moment-matching is performed. In view of implementation, we provide algorithms that identify the required Krylov subspaces without explicitly computing the projectors used in the analysis. The benefits of our approach are illustrated for the quadratic-bilinear descriptor systems corresponding to semi-discretized Navier–Stokes equations.

Stability of weakly damped MDGKN-systems: The role of velocity proportional terms

Abstract: Many problems in mechanical engineering, in the linearized case, can be modeled as a second order system of differential equations of type MDGKN, where the matrices correspond to inertia, damping, gyroscopic, stiffness, and circulatory forces. The latter may lead to self-excited vibrations which in general are unwanted and sometimes dangerous. It is well known that circulatory systems are very sensitive to damping and their stability behavior may strongly depend on the structure of the damping matrix. Moreover, it has been known for a long time, that the addition of (even infinitesimally small) damping may also destabilize such systems. The present note studies in more detail some of the effects of velocity proportional terms on the stability of mechanical systems of this type. The aim is to extend the findings recently presented by HAGEDORN et al. Therefore, the analytically derived stability boundary of an MDGKN-system with two degrees of freedom is analyzed with regard to infinitesimally small, incomplete, and indefinite damping matrices as well as the role of gyroscopic terms and the spacing of the eigenfrequencies.

Geometry of principal stress trajectories for a Mohr-Coulomb material under plane strain

Abstract: In the mechanics of granular and other materials the system of equations comprising the Mohr-Coulomb yield criterion together with the stress equilibrium equations under plane strain conditions forms a statically determinate system. The results presented here for this system are consequently independent of any flow rule that may be chosen to calculate the deformation and also independent of whether elastic strains are included. The stress equilibrium equations are written relative to a coordinate system in which the coordinate curves coincide with the trajectories of the principal stress directions. The general solution of the system is constructed relative to this coordinate system. A simple relation connecting the two scale factors for the coordinate curves is derived. In the special case that the angle of internal friction vanishes, this relation reduces to that already available for the Tresca yield criterion. A simple example is presented to illustrate the general solution. Finally, a boundary value problem for the region adjacent to an external boundary which coincides with a principal stress trajectory is formulated and its solution is outlined.

The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D

Abstract: In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field $F := abla\varphi: \Omega \to \mathrm{GL}^+(n)$ and the microrotation field $R: \Omega \to \mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form \begin{equation*} W_{\mu,\mu_c}(R\,;F) := \mu\,\left\lVert{\mathrm{sym}(R^T F - \boldsymbol{1})}\right\rVert^2 + \mu_c\,\left\lVert{\mathrm{skew}(R^T F - \boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $\mu > 0$ is the Lame shear modulus and $\mu_c \geq 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations $\mathrm{argmin}_{R\,\in\,\mathrm{SO}(n)}{W_{\mu,\mu_c}(R\,;F)}$ as a function of $F \in \mathrm{GL}^+(n)$ and weights $\mu > 0$ and $\mu_c \geq 0$. For $n \geq 2$, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: $(\mu, \mu_c) = (1,1)$ and $(\mu,\mu_c) = (1,0)$. In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range $\mu > \mu_c \geq 0$ in dimension $n\!=\!2$. Currently, optimality results for $n \geq 3$ are out of reach for us, but we contribute explicit representations for $n\!=\!2$ which we name $\mathrm{rpolar}^{\pm}_{\mu,\mu_c}(F) \in \mathrm{SO}(2)$ and which arise for $n\!=\!3$ by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations $\mathrm{rpolar}^\pm_{\mu,\mu_c}(F_\gamma)$ for simple planar shear.

On a method for solving Prandtl's integro-differential equation applied to problems of continuum mechanics using polynomial approximations

Abstract: In the present paper, using the ideas of the well-known method for solving singular integral equations (SIE), based on Gaussian quadrature formulas for ordinary integrals and singular integral with Cauchy kernel, a somewhat different approach is proposed for solving the Prandtl integro-differential equation (IDE). This approach in accordance with a polynomial method includes different approximation methods for a continuous component of Prandtl's IDE solution, namely an approximation in the form of Bernstein's polynomials, Chebyshev's polynomials of the first kind and approximation in the form of Lagrange interpolation polynomial at the Chebyshev knots. As a result, the Prandtl IDE is reduced to the system of linear algebraic equations (SLAE) of the rather simple structure. These different approximation methods will be illustrated by the example of the problem on contact interaction between a stringer of finite length, having a variable along the length rigidity in tension-compression or a variable cross-section, and an elastic semi-infinite plate. This interaction is described by the Prandtl IDE. The comparative numerical analysis of the results obtained by different methods is carried out.

Torsion analysis of finite solid circular cylinders with multiple concentric planar cracks

Abstract: First, using the separation of variables technique a series solution for torsion of an intact finite cylinder under self-equilibrating lateral shear tractions is given. Next, an integral form solution for torsion of an intact infinite cylinder under the aforementioned lateral loading is obtained. After that the solution of an axisymmetric rotational Somigliana ring dislocation in the infinite circular cylinder is obtained. The distributed dislocation technique is used to construct integral equations for stress analysis of the infinite cylinder with a set of coaxial axisymmetric cracks. These defects are penny-shaped, annular and circumferential edge cracks. The infinite cylinder is under the action of the loading similar to the intact cylinder. The integral equations are solved numerically to obtain the dislocation density on the surfaces of the cracks. The stress intensity factors for the cracks are determined by employing the dislocation densities. The problem of a cracked finite cylinder is treated by cutting method. That is, a similar cracked infinite cylinder is sliced by extending two additional annular axisymmetric cracks. To validate the cutting method, the solution of the sliced intact infinite cylinder is compared to that of an intact finite cylinder which both of them are under the same self-equilibrating lateral shear tractions. The solution is applied to several examples to study the effect of crack type/location on the resulting stress intensity factors at tips of the cracks.

A comparison of explicit and semi‐implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime

Abstract: This paper compares the accuracy and computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric compliant tubes. There are essentially two main classes of mathematical models that can be used to predict the pressure and velocity distribution along the tube: one class is based on the full compressible Navier-Stokes equations in an axially symmetric geometry, leading to a two-dimensional governing PDE system with moving boundaries, and the other class uses a simpler, cross-sectionally averaged version of the Navier-Stokes equations, which leads to a non-conservative PDE system in only one space dimension along the axial direction of the tube. Within the first class of models, the influence of the wall friction on the flow field is directly obtained from first principles, without any further modelling assumptions and is thus valid even for highly unsteady flows. In the second case, only averaged flow quantities are available, and it is well known from previous studies published in the literature that the correct representation of the wall friction needs to be frequency dependent, since the use of a simple steady friction model, like the classical Darcy-Weisbach law, is not sufficient to reproduce the wall friction effects in highly transient flows. For the cross-sectionally averaged Navier-Stokes equations, there are again two main classes of frequency-dependent wall friction models: convolution integral (CI) models and instantaneous acceleration (IA) models. In this paper we provide a very thorough and critical comparison of all the above-mentioned methods for the simulation of highly oscillatory flows in rigid and compliant tubes concerning accuracy and computational efficiency. From our numerical results we can conclude that the convolution integral models are significantly superior to instantaneous acceleration models concerning accuracy. Furthermore, the CI models require only a slight computational overhead if they are properly implemented by solving a set of additional ODEs for appropriate auxiliary variables, instead of directly computing the convolution integrals. We also find that semi-implicit finite volume methods are clearly superior to conventional explicit finite volume schemes concerning computational efficiency, however, providing the same level of accuracy.

Elastoplastic analysis of pressure-sensitive materials by an effective three-dimensional mixed finite element

Abstract: This work is focused on the analysis of three-dimensional bodies whose mechanical behavior can be modeled by an elastoplastic pressure-sensitive material description. To this end the yield condition is assessed with respect to a general quadratic function capable to represent both standard surfaces, such as Drucker–Prager surface, or more generic surfaces. For the sake of the efficiency and robustness of the numerical analysis of general-shape solids, an effective mixed tetrahedral finite element is used to perform a step-by-step analysis obtaining the complete equilibrium path and the collapse load. Some numerical results regarding technical problems, such as masonry walls and footing problems, show the effectiveness of the mechanical model and of the numerical strategy.