VDI Verlag eLibrary

FE-Modellierung von Elastomerkomponenten mit textilen Verstärkungscorden am Beispiel von Luftfedern

Nina Heinrich

Series: Mechanik/Bruchmechanik

Volume 358 2022

Results in this title

Mathematische und mechanische Grundlagen S. 14-25

„… TensorrechnungTensoren sind gerichtete physikalische oder geometrische Größen, die sich aus Koeffizi-enten und einer …” „… Koeffizienten von Tensoren höherer Stufe lassen sich in3.1 Tensorrechnung 15Matrixform darstellen:[va] =vxvyvz …” „… ◦ eb ◦ ea →3J ··X = X ··3J = X , (3.27)können zwei Tensoren,SJ =12(2J +3J) →SJ ··X = X ··SJ = XS …”

Titelei/Inhaltsverzeichnis S. I-XII

„… −1 inverser TensorXT transponierter TensorXS, XA symmetrischer und antimetrischer AnteilXH , XD …” „… InterpolationsintervallsAbkürzungs- und Symbolverzeichnis IXI1, I2, I3 Hauptinvarianten eines zweistufigen TensorsĪ1, Ī2, Ī3 …” „… Verschiebung des Knotens KTensorenA Strukturtensorb linker Cauchy-Green-TensorC rechter Cauchy-Green-Tensorε …”

Zusammenfassung und Ausblick S. 144-164

„… Variablesymmetrische Tensoren zweiter Stufe sind. Für den Spezialfall einer skalarwertigenTensorfunktion Y = Y (X …” „… Schläuche.149A AnhangA.1 Zusammenhänge zwischen Hencky-Tensor und StrecktensorUm einen direkten Bezug zu …” „… Hauptinvariante des Euler’schenHencky-Tensors dem natürlichen Logarithmus der Volumenstreckung entspricht:I1(∆h …”

Literatur S. 165-174

„… Kon-tinuumsmechanik. Bd. 340. Düsseldorf: VDI-Verlag.Itskov, M. (2015). Tensor algebra and tensor analysis for …”

Materialbeschreibung S. 50-63

„… = ln((∆V)). (5.6)Es handelt sich also um den inkrementellen Euler’schen Hencky-Tensor.5.3 Filamentmaterial …” „… derCauchy-Green-Tensoren (siehe Gleichung (3.58)). Der von Hartmann & Neff (2003)vorgeschlagene Ansatz für den …” „… Tensoren ebenfalls aus der Summe der einzelnen Anteile:τ = τani + τ iso + τ vol , (5.16)k = kani + kiso …”

Geometrische Modellierung textiler Corde S. 26-49

„… )]=∂ AXC∂ϕ̄eC . (4.39)Bei der in Gleichung (4.36) auftretenden Ableitung der Koeffizienten des Rotations-tensors …” „… Gleichungen hier noch einmal wiedergege-ben, wobei die tensorielle Darstellung teilweise leicht abgewandelt …”

Zyklisch symmetrische Balgwandmodelle S. 64-102

„… und verschiedenste Invarianten dieser Tensoren. Ein Beispiel enthältAbbildung 6.28 mit der Verteilung …”

Grundlagen und Inhalte der Arbeit S. 3-13

„… Arbeit relevanten Aspekte derTensorrechnung sowie der nichtlinearen Kontinuumsmechanik und FEM.Das …”

doi.org/10.51202/9783186358189

No access

Select for citation download

Improving the robustness of the finite cell method for problems with large deformations

Wadhah Garhuom

Series: Mechanik/Bruchmechanik

Volume 359 2024

Results in this title

Basic equations of continuum mechanics S. 6-16

„… important quantity is the deformation gra-dient tensorF = Gradφ(X, t) =dxdX. (2.2)This describes how an …” „… relationship between the Cauchy stress tensor σ and the engineering strain tensorε as followsσ = 2µ ε+ λ trε1 …” „… fourth-order material tangent tensorA =∂P∂F. (2.66)Finally, substituting (2.65) and (2.57) in (2.64) results in …”

Remeshing strategy for large deformations in the FCM S. 89-129

„… gradient being equal to the null tensor. If thedisplacement gradients are interpolated successfully, all …” „… inverseplastic right Cauchy-Green strain tensor C−1p (2.52) and the equivalent plastic strain ᾱ(2.51), see …” „… Section 2.3.3 for more details. Rather than interpolating the non-smooth plasticstrain tensor C−1p to the …”

Non-negative moment fitting quadrature for cut finite elements and cells S. 27-58

„… . This can beeasily obtained by applying a tensor product of the one-dimensional polynomial functions.For …”

doi.org/10.51202/9783186359186

No access

Select for citation download

Partitioned solution strategies for electro-thermo-mechanical problems applied to the field-assisted sintering technology

Patrick Erbts

Series: Mechanik/Bruchmechanik

Volume 345 2016

Results in this title

Titelei/Inhaltsverzeichnis S. I-XVI

„… the following. Further reading on tensor-algebra and tensor-analysis for continuum mechanics is …” „… 186 VIII Nomenclature A distinction is made in scalars, vectors, tensors, and matrices and the …” „… are indicated by arrows a, second-order tensors are written with an under-tilde ∼S. This holds also …”

Governing equations S. 8-44

„… that is of interest here. It can be computed by means of the so-called deformation gradient tensor ∼F …” „… split into two parts ∼F = ∼R∼U = ∼V∼R , (2.7) where ∼R is the rotation tensor, ∼U the right stretch …” „… tensor related to the undeformed reference configuration, and ∼V is the left stretch tensor defined in …”

Numerical methods S. 45-78

„… the first Piola-Kirchhoff stress tensor ∼P. In order to obtain a description that is formulated in …” „… terms of the symmetric second Piola-Kirchhoff stress tensor, the relation ∼P = ∼F∼S is used. Following …” „… ) = ∼S · δ∼E , (3.7) where δ∼E is the virtual Green-Lagrange strain tensor. This finally leads to the …”

Numerical examples S. 145-170

„… constitutive model and material parameters are utilized. Concerning the kinematic relations, the strain tensor …” „… function. Based on this function, the expressions for the stress tensor and the entropy can be calculated …”

Solution of coupled problems S. 79-144

doi.org/10.51202/9783186345189

Partial access

Select for citation download

The hierarchical finite cell method for nonlinear problems: Moment fitting quadratures, basis function removel, and remeshing

Simeon Hubrich

Series: Mechanik/Bruchmechanik

Volume 355 2021

Results in this title

Basic elements of continuum mechanics S. 6-22

„… 1 denotes the second-order identity tensor. Now, having introduced the deformation gradient allows …” „… deformation gradient multiplicatively into a rotation and a stretch tensor where the stretch characterizes the …” „… decomposition of the deformation gradient F = RU = VR , (2.17) where R denotes the rotation tensor, which is …”

The finite cell method S. 23-35

„… the Cauchy stress tensor, we obtain∫ ϕ(Ωe) α (a grad ∆u) · grad η dv = ∫ ϕ(Ωe) α ( ρb · η dv − σ · ∇S …” „… stated in the standard matrix notation. Therefore, in order to distinguish between matrix and tensor …” „… tensor product of the one-dimensional quadrature points. In the following, this is demonstrated for the …”

A remeshing strategy for the FCM S. 124-149

„… them from tensor quantities. Next, we evaluate the remeshing criteria based on the updated solution Ũi …” „… quantities in order to distinguish them from tensor ones. Finally, the criteria may be evaluated for …”

Moment fitting quadratures S. 36-78

„… tensor-product of the onedimensional Legendre polynomials as F = {Lu(ξ)Lv(η)Lw(ζ), u, v, w = 0, . . . , pq …” „… basis functions ϕj (ξ) are constructed applying a tensor product of the pq +1 one-dimensional Lagrange …” „… polynomials, and the corresponding points ξi are obtained by the tensor product grid of the pq + 1 …”

doi.org/10.51202/9783186355188

Partial access

Select for citation download

Experimentell-numerische Vorgehensweise zur Entwicklung von Probekörper-Setups für die Charakterisierung technischer Elastomere

Lars Kanzenbach

Series: Mechanik/Bruchmechanik

Volume 357 2021

Results in this title

Grundlagen S. 5-17

„… Tensoralgebra Die im Rahmen dieser Arbeit benötigten Vektoren und Tensoren werden durch Unterstriche pro Stufe …” „… aus einem TensorX Invarianten berechnen. Die etablierten Hauptinvarianten I1, I2 und I3 sind wie folgt …” „… gekennzeichnet. Beispielsweise stellt der Tensor X einen Tensor 2. Stufe dar. Dieser setzt sich aus dem …”

Titelei/Inhaltsverzeichnis S. I-XII

„… Tensorkoeffizienten [Xab ] Koeffizientenmatrix X Tensor nullter Stufe (Skalar) XA Eigenwerte eines Tensors X̃ Größe in …” „… Größen X, X Tensor erster und zweiter Stufe XT , X 1 Transponierter und inverser Tensor S X, A X …” „… Symmetrischer und antisymmetrischer Anteil eines Tensors h X, X ′ Hydrostatischer und deviatorischer Anteil …”

Probekörperdesign für hochpräzise Zug-/Druckmessungen S. 18-87

„… der Grundlage tensorieller Größen und besitzt den Vorteil, dass selbst mehrachsige und inhomogene …”

doi.org/10.51202/9783186357182

Partial access

Select for citation download

Partitioned Solution Strategies for Strongly-Coupled Fluid-Structure Interaction Problems in Maritime Applications

Marcel König

Series: Mechanik/Bruchmechanik

Volume 351 2018

Results in this title

Titelei/Inhaltsverzeichnis S. I-XVI

„… -, vector-, or tensor-valued quantity u Generic discrete quantity U Dilatational part of the strain energy …” „… In this thesis, the following notation for the distinction of scalars, vectors, tensors, and matrices …” „… second-order tensors T defined as linear maps between vector spaces are typeset in bold italic font. Bold …”

Structural Problems S. 25-41

„… second-order identity tensor by I and recalling the definition of the deformation gradient F in Equation (3.2 …” „… be decomposed into an orthogonal rotation tensor R and a symmetric stretch tensor U or V …” „… , respectively, such that F = RU = V R . (3.11) The right stretch tensor U and the left stretch tensor V are …”

Fluid Problems S. 5-24

„… this equation is still retained if φ is a vector- or tensor-valued quantity. Equation (2.27) represents …” „… , (2.44) where σ denotes the stress tensor and b represents the volume force per unit mass. Recalling …” „… ) + ρbi dΩ , (2.55) or, again in tensor notation, ∫ Ωt ρ ⎛ ⎝ ∂v ∂t ∣∣∣∣∣ χ + ṽ · gradv ⎞ ⎠ dΩ = ∫ Ωt divσ …”

Software Library comana S. 98-157

„… array and tensor types. Eigen makes heavy use of expression templates and generates expression trees at …” „… integration points. In the getIntegrationPoints method, the integration points are initialized as the tensor …”

doi.org/10.51202/9783186351180

Partial access

Select for citation download

Expressive Personalized 3D Face Models from 3D Face Scans

Stella Graßhof

Series: Informatik/ Kommunikation

Volume 868 2020

Results in this title

Experiments S. 146-173

„… versions approximate the original data tensor, using the same cropping factors, i.e. dimension one for the …” „… level. Given the original 3D face shapes as data tensor T4,0 ∈ R3N×P ×L×E , and T 4,apathy representing …” „… the tensor which consists of the mean apathetic face, while T 4,neutral contains the mean neutral face …”

Fundamentals S. 15-39

„… defined as a tensor product based on applying the traditional SVD on the individual tensor-unfoldings, as …” „… the core tensor accordingly, see Fig. 2.2 for illustration. As the presented tensor-factorization is …” „… . 2.3 Factorization 25 Fourth-Order Cumulant Tensor The previously introduced fourth-order cumulants are …”

Face Models S. 114-145

„… , followed by the proposed tensor-based face model in different variants. An analysis of the proposed model …” „… ) is commonly used in published face tensor-models on both image and 3D data [18, 47, 30, 66] and is an …” „… position change. This process results in a fine detailed 3D face reconstruction. 5.3 Proposed Tensor Face …”

Summary and Conclusions S. 174-178

„… aligned and the resulting balanced data can be sorted into a data tensor, which is used in the following …” „… . In Chapter 5 four tensor face models, build from three different databases were presented, along with …” „… usages. Afterwards in order to investigate effects of the parameterizations of the tensor models, they …”

Titelei/Inhaltsverzeichnis S. I-XIV

„… – correspondence estimation – expression intensity – tensor – factorization – statistical models – expression …” „… – Korrespondenzschätzung – Intensität von Gesichtsausdrücken – Tensor – statistische Modelle – Transfer von Ge …” „… Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Notation …”

Index S. 199-204

„… Animation Parameters, see FAP FACS, 5 FAP, 5 ffp, 42 flattening of tensor, 28 fooling, 143 Gaussian …” „… , 68 Singular Value Decomposition, see SVD SVD, 22, 30 tensor, 27, 116 time alignment, 100 transfer …” „… , 149 expression, 150 person, 150 uncanny valley, 2 unfolding of tensor, 29 whitening, 22 Wolfe …”

Introduction S. 1-14

„… from 2D input the proposed tensor-based face models outperform state-of-the-art approaches. 1.5 Thesis …” „… factorization approach of a higher-dimensional data structure, i.e. data tensor, of processed 3D face scans. The …” „… synchrony of facial motion. Chapter 5: Face Models • The factorization of 3D face tensors reveals structures …”

Appendix S. 179-188

„… information [75]. Columns five to seven show results of the proposed tensor models build from three different …” „… , whereas all tensor models rely on a subset of the shown 2D landmarks only. The presented examples confirm …”

Literature S. 189-198

„… Ensembles: TensorFaces. In: European Conference on Computer Vision (ECCV). 2350. 2002, pp. 447–460 …” „… : 10.1002/0471221317. [45] J. F. Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the …”

From 3D Face Scans to Aligned Faces S. 59-113

„… points, because otherwise they cannot be ordered into a matrix or tensor. 6. The points between each pair …” „… sequence s can be ordered into a data tensor Fs ∈ R3×N×Ts , s = 1, . . . , S. If the number of frames T …” „… statistical model [66, 30, 70, 71]. To reach this goal, the dimensionality of each of the 3D data tensors Fs …”

doi.org/10.51202/9783186868107

Partial access

Select for citation download

A partitioned solution approach for fluid-structure interaction problems in the arterial system

Lars Radtke

Series: Mechanik/Bruchmechanik

Volume 353 2020

Results in this title

Appendix S. 257-268

„… tensors A and B, and fourth-order tensorsA and B, the following operations are defined according to the …” „… Appendix A.1 Tensor algebra For an in-depth treatment of tensor algebra as well as tensor analyisis …” „… literature. A.1.1 Contractions and scalar products Considering first-order tensors a and b, second-oder …”

Mechanical modeling of the arterial system S. 23-68

„… spatial flux tensor Γ(x) and source vector β, one obtains∫ Ω(x) ∂ω ∂t ∣∣∣∣ x + div ( ω ⊗ v − Γ(x) ) − β …” „… form, respectively. From (3.41), the transformation rule for the flux tensor Γ(X) = J Γ(x)F−T (3.43 …” „… momentum, the Cauchy stress tensor σ can be identified as the spatial flux of linear momentum and sources …”

Titelei/Inhaltsverzeichnis S. I-X

„… . . . . . . . . . . . . . . . . . . . . . . . . 251 7 Summary and Outlook 253 Appendix 257 A.1 Tensor algebra …” „… order tensors . . . . . . . 257 A.1.3 Special fourth order tensors . . . . . . . . . . . . . . 258 A.2 …” „… Continuum mechanics . . . . . . . . . . . . . . . . . . . . . 258 A.2.1 Neo-Hookean elasticity tensor …”

Numerical methods S. 69-152

„… polynomial tensor-product basis of degree pg = 4 is considered. This gives 25 degrees of freedom for each …” „… = DdE∆d denotes the directional derivative of the Green- Lagrange strain tensor. Note that for the dynamic …” „… (4.30), CV denotes the Voigt representation of the elasticity tensor C = ∂S/∂E (see Section A.3.4 for …”

Numerical investigations S. 153-219

„… well. Therein, the trunk and the tensor product space were considered, while this study is augmented by …” „… ) is used instead of the tensor product space (TPS), the same error can be achieved with less degrees …” „… tensors, a corresponding addition is obtained as explained in Section 3.2.2. As can be seen from Figure …”

Bibliography S. 269-288

„… , 2010, pp. 143–178. [48] R. de Boer. Vektor- und Tensorrechnung für Ingenieure. Berlin and Heidelberg …” „… . Weller, G. Tabor, H. Jasak, and C. Fureby. “A Tensorial Approach to Computational Continuum Mechanics …”

doi.org/10.51202/9783186353184

Partial access

Select for citation download

Bauingenieur

Browse Volumes: Bauingenieur

Bauingenieur Volume 94 (2019) Issue 12 ISSN Online: 0005-6650

Results in this title

Mehran Motevalli, Jörg Uhlemann, Natalie Stranghöner, Daniel Balzani A new nonlinear polyconvex orthotropic material model for the robust simulation of technical fabrics in civil engineering applications at large strains – Validation with large-scale experiment/Ein neues polykonvexes orthotropes Materialmodell zur robusten Simulation von Textilmembranen im Bauingenieur- wesen unter Berücksichtigung großer Deformationen – Validierung anhand eines Großbauteilversuchs S. 488-497

„… simulation of tensile membrane structures. The notion of anisotropic metric tensors is employed in the …” „… interactions, one may directly use multiplicative mixed invariant terms of the right Cauchy-Green tensor and …” „… the classical structural tensors [13], [14]. However, these non-polyconvex formulations may lead to …”

doi.org/10.37544/0005-6650-2019-12

No access

Select for citation download

Partitioned Fluid-Structure Interaction Simulation of Maritime Applications

Jorrid Lund

Series: Mechanik/Bruchmechanik

Volume 360 2024

Results in this title

Structural mechanics S. 5-20

„… vectorand tensor-based quantities are written in bold to distinguish them from scalar-valuedquantities.Based …” „… )holds. C is the right Cauchy-Green tensor that relates to the Green-Lagrange strain tensoras C = 2 E + I …” „… tensor σ containsthe stress components σij. The body force vector b captures volumetric loads. All …”

Fluid mechanics S. 21-34

„… )In Eq. (3.4), p is the pressure, µ is the dynamic viscosity, and I the identity tensor.To obtain the …” „… divergence of the fluid stress tensorT (defined in Eq. (3.4)) in Eq. (3.5) is reduced to div(T ) = −div(pI …” „… tensor.To obtain the incompressible momentum equation, Eq. (3.3) is divided by the density ϱwhich is assumed …”

Introduction S. 1-4

„… vector b captures volumetric loads. All vectorand tensor-based quantities are written in bold to …” „… the displacement vector). The Cauchy stress tensor σ containsthe stress components σij. The body force …” „… the symmetry of theCauchy stress tensor σ [31, p. 104].If the deformation of a body is large, it is …”

Submersible mixer S. 71-80

„… to use the engineering strain tensor to compute the von Misesequivalent strain. Since the strain in …” „… to use the engineering strain tensor to compute the von Misesequivalent strain. Since the strain in …”

Bibliography S. 124-138

„… , release 19.2 (2018).[75] H. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to …” „… , release 19.2 (2018).[75] H. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to …”

doi.org/10.51202/9783186360182

No access

Select for citation download

Numerische Homogenisierung von Faserverbundwerkstoffen mit periodischer Mikrostruktur

Mathias Würkner

Series: Mechanik/Bruchmechanik

Volume 346 2017

Results in this title

Modellierung von Werkstoffen S. 9-20

„… Proportionalitätsfaktoren Cijkl sind Komponenten eines Tensors C vierter Stufe, die infolge der Homogenität des betrachteten …” „… Körpers invariant bezüglich x und somit konstant sind. Dieser Tensor wird Elastizitätstensor genannt …” „… durch diese Koeffizienten definierte Tensor vierter Stufe wird Nachgiebigkeitstensor genannt. Aus der …”

Titelei/Inhaltsverzeichnis S. I-XVI

„… Verschiebung E Vektor der elektrischen Feldstärke e Tensor der piezoelektrischen Kostanten (3. Stufe) κ Tensor …” „… bezüglich φ ẽ Matrixschreibweise des Tensors der piezoelektrischen Konstanten φ̂ vorgeschriebenes …” „… piezoelektrischen Tensors in Ω f emkij , i, j, k = 1, 2, 3 Koeffizienten des piezoelektrischen Tensors in Ω m κfij …”

Homogenisierungsverfahren S. 21-52

„… Koeffizienten materialbeschreibender Tensoren, wie zum Beispiel den Elastizitätstensor C und den Tensor der …” „… ) wobei die Komponenten ε0ij eines Tensors ε 0 vorgegeben sind. Die erste Bedingung aus Gl. (3-21) kann …” „… ) werden sechs Probleme mit speziellen vorgegebenen symmetrischen Tensoren ε0 betrachtet. Diese Tensoren …”

Einleitung S. 1-8

„… von materialbeschreibenden Tensoren eingegangen. Das dritte Kapitel widmet sich der …”

Berechnung effektiver Materialeigenschaften S. 65-112

„… Materialeigenschaften mation auf globale Koordinaten verändert sich die Darstellung des Tensors zu Gl. (2-43). Das …” „… Koordinatensystem ergibt sich ein Elastizitätstensor, dessen Darstellung in M-V-Notation der des Tensors bezüglich …” „… Transformation in ein geeignetes lokales Koordinatensystem kann der Tensor in M-V-Notation die Darstellung aus Gl …”

doi.org/10.51202/9783186346186

Partial access

Select for citation download

International Conference on Gears 2019

VDI Wissensforum GmbH (ed.)

Series: VDI-Berichte

Volume 2355 2019

Results in this title

C. Brecher, C. Lopenhaus, F. Goergen Transfer of the tooth flank stress into an analogy test concept to provoke the damage pattern tooth flank fracture (TFF) S. 25-36

„… determining the optimal position of the primary actuator. The results show that comparable stress tensor …” „… loads only maps the maximum amplitudes of the normal stresses precisely, but not the stress tensor …” „… , deviations in the stress tensor sequence became visible, mainly in the case of tangential compressive …”

N. C. Ecke, A. K. Schlarb, J. Holler, Jan Niedermeyer, P. Klein Thermal simulation of anisotropic PEEK-based tribocomposites S. 1411-1420

„… homogenization, the fiber orientation tensor a defined by Advani and Tucker [16] was determined from the …” „… orientation tensor diagonal element: ,∥ ∙ , ∙ 1 Numerical homogenization was performed by determining an …” „… , C.L.: The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites. J Rheol …”

J.-A. Meis, F. Teklote, M. Borowski Simulation of the tooth root strength under consideration of material quality, finishing process and size effects S. 161-172

„… leading to premature contact. This induces the maximum stresses in the tooth root. The stress tensor and …” „… amplitude σA are derived from the stress tensor for each element of the FE-model. Then σM and σA are …”

B. H. Kien, D. Iba, Y. Ishii, Y. Tsutsui, N. Miura, T. Iizuka, A. Masuda, A. Sone, I. Moriwaki Vibration-based plastic gear crack detection system using a pre-learned convolutional neural network S. 1443-1452

„… as TensorFlow and Keras, are available to employ pre-trained CNNs in an efficient way. Instead of …”

doi.org/10.51202/9783181023556

Partial access

Select for citation download

International Conference on Gears 2017

VDI Wissensforum GmbH (ed.)

Series: VDI-Berichte

Volume 2294 2017

Results in this title

B. Schlecht, C. Ulrich, J. Bräunig Investigations on the influence of anisotropic material properties of gear teeth on their macroscopic behavior S. 659-668

„… anisotropic stiffness- and compliance-tensor for a material area is not trivial. This chapter briefly …” „… compliance for material areas As discussed in detail by [3], the mean stiffness and compliance tensor for a …” „… . 2294, 2017 661 where and are the second-order stress and strain tensors, is the fourth-order …”

C. Brecher, C. Löpenhaus, F. Goergen, D. Mevissen Crack propagation analysis of pitting damages of high-strength material systems in gear applications S. 281-292

„… stress tensors. In addition to the equivalent stresses according to the shape modification hypothesis …” „… stress tensor in the XY plane in the tooth center. The principal stresses in lead direction are not …” „… calculate the principal stresses and principal stress directions. The stress tensor, consisting of two …”

C. Brecher, C. Löpenhaus, J. Pollaschek Influence of surface layer properties on tooth root bending strength of cylindrical gears S. 421-432

„… MURAKAMI’S work. The calculation method is based on the stress tensors in the tooth root area of the …” „… from the stress tensors is compared to the local fatigue strength of every defect inside and on the …”

doi.org/10.51202/9783181022948

Partial access

Select for citation download

PIAE EUROPE 2021

VDI Wissensforum GmbH (ed.)

Series: VDI-Berichte

Volume 2385 2021

Results in this title

Daniel Kugele, Sebastan Mönnich, Cigdem Belgeli Development and validation of a simulation methodology for the prediction of surface defects for plastic components with metallic effect pigments S. 89-100

„… second-order tensors to describe orientation distributions [2], two model parameters are available to the user …” „… of the Moldflow simulation is, as already mentioned, a second-order orientation tensor, which …” „… flakes and surface normals is to be determined from the secondorder orientation tensor is described in …”

Daniel Kugele, Sebastan Mönnich, Cigdem Belgeli Entwicklung und Validierung einer Simulationsmethodik zur Vorhersage von Oberflächendefekten für Kunststoffbauteile mit metallischen Effektpigmenten S. 101-112

„… Themengebiet auf: Beginnend mit dem Folgar-Tucker Modell (FT) [1] und der Einführung von Tensoren 2ter Stufe …” „… [2] beschrieben. Es kann zusammengefasst abgeleitet werden, dass die Eigenwerte des Tensors den …” „… . Advani, S. und Tucker, C. The Use of Tensors to Describeand Predict Fiber Orientation in Short Fiber …”

S. Jana, F. Ehrhart, M. Bromberger The History of Integrative Simulation from Laboratories to all Designers’ Desktops S. 113-128

„… short-fiber reinforced material, by combining the local fiberorientation tensor with a microscale based …” „… fiberorientation tensor: a_xx and a_zz in global coordinate system source: Altair Efficient Material Modelling …”

doi.org/10.51202/9783181023853

Partial access

Select for citation download

The hierarchical finite cell method for problems in structural mechanics

Meysam Joulaian

Series: Mechanik/Bruchmechanik

Volume 348 2017

Results in this title

Titelei/Inhaltsverzeichnis S. I-X

„… they appear for the first time. Tensors A, a Scalar value a = ai ei First order tensor (vector) ∼S …” „… = Sij ei ⊗ ej Second order tensor ∼C = Cijkl ei ⊗ ej ⊗ ek ⊗ el Fourth order tensors Matrices a …” „… Mathematical tensor operations u⊗ v = ui vj ei ⊗ ej Dyadic product ∼S · ∼F = Sij Fij Inner, scalar or dot …”

Summary and Outlook S. 133-135

„… cell method, the shape functions are based on the tensor-product of Lagrange polynomials of order p …” „… employed a tensor-product of the 1D Gauss-Legendre-Lobatto quadrature rule with p + 1 Gauss points. Due to …”

Finite cell method for problems in solid mechanics S. 5-26

„… × ] 0, t̂ [ (2.1) 6 2 Finite cell method for problems in solid mechanics where ∼σ is the stress tensor …” „… = ∼C∼ε ∼ε = 1 2 ( grad u+ gradT u ) (2.2) where∼ε is the strain tensor and ∼C is the fourth-order …” „… elasticity tensor. These equations are defined along with the following initial conditions, u( x, 0) = u0 …”

The spectral cell method S. 111-132

„… (7ξ 2 − 3)(ξ + 1) 2D and 3D shape functions can be obtained by applying the tensor-product of the 1D …” „… a mesh, and applying a tensor-product space of order p = 3, the model contains almost nD ≈ 6.5 × 105 …”

Numerical integration algorithms for the FCM S. 27-71

„… tensor-product of the one-dimensional Gaussian quadrature as ∫ P(x(ξ)) detJ c d = nζg∑ k=1 nηg∑ j=1 nξg∑ i=1 …” „… . quadrature at hand. In this thesis, we select the basis function based on the tensor-product of the …” „… tensor-product, the following relation between the order of the quadrature pq and the number of the basis …”

Local enrichment of the FCM S. 72-110

„… − ξj ξi − ξj , i = 1, 2, . . . , pm + 1. (4.12) 2For 2D and 3D, a tensor-product rule of 1D …” „… tensor-product of the one-dimensional functions [25, 93]. The accuracy of such an interpolation can be increased …”

doi.org/10.51202/9783186348180

Partial access

Select for citation download

Results (143)