„… 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 …”

„… 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 …”

„… 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 …”

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

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

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

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

„… 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 …”

„… × ] 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 …”

„… (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 …”

„… 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 …”

„… − 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 …”