Tensors

Scientia Iranica, 2007

In this paper, a uni ed explicit tensorial relation is sought between two stress tensors conjugate to arbitrary and general Hill strains. The approach used for deriving the tensorial relation is based on the eigenprojection method. The result is, indeed, a generalization of the relations that were derived by Farahani and Naghadabadi [1] in 2003 from a component to intrinsic form. The result is uni ed in the sense that it is valid for all cases of distinct and coalescent principal stretches. Also, in the case of three dimensional Euclidean inner product space, using the derived uni ed relation, some expressions for the conjugate stress tensors are presented.

Journal of Elasticity - J ELAST, 2000

Let E be a 3-dimensional Euclidean space, and let V be the vector space associated with E. We distinguish a point p ∈ E both from its position vector p(p) := (p − o) ∈ V with respect to a chosen origin o ∈ E and from any triplet (ξ 1 , ξ 2 , ξ 3 ) ∈ IR 3 of coordinates that we may use to label p. Moreover, we endow V with the usual inner product structure, and orient it in one of the two possible manners. It then makes sense to consider the inner product a · b and the cross product a × b of two elements a, b ∈ V; in particular, we define the length of a vector a to be |a| = (a · a) 1/2 , and denote by U := {v ∈ V | |v| = 1} the sphere of all vectors having unit length. When needed or simply convenient, we think of E as equipped with a Cartesian frame {o; c 1 , c 2 , c 3 } with orthogonal basis vectors c i ∈ U (i = 1, 2, 3); the Cartesian components of a vector v ∈ V are then v i := v · c i and, in particular, the triplet (p 1 , p 2 , p 3 ) ∈ IR 3 , p i := p(p) · c i , of components of the position vector are the Cartesian coordinates of a point p ∈ E.

Mechanics Research Communications, 2012

A stress gradient continuum theory is presented that fundamentally differs from the well-established strain gradient model. It is based on the assumption that the deviatoric part of the gradient of the Cauchy stress tensor can contribute to the free energy density of solid materials. It requires the introduction of so-called micro-displacement degrees of freedom in addition to the usual displacement components. An isotropic linear elasticity theory is worked out for two-dimensional stress gradient media. The analytical solution of a simple boundary value problem illustrates the essential differences between stress and strain gradient models.

International Journal of Solids and Structures, 2003

In this paper, general relations between two different stress tensors T f and T g , respectively conjugate to strain measure tensors f ðUÞ and gðUÞ are found. The strain class f ðUÞ is based on the right stretch tensor U which includes the Seth-Hill strain tensors. The method is based on the definition of energy conjugacy and HillÕs principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth-Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T ð0Þ which is the stress conjugate to the logarithmic strain tensor lnU.

In this work, the skew-symmetric character of the couple-stress tensor is established as the result of symmetry arguments associated with the pseudo character of the couple-traction vector. We notice that while the normal component of this vector on any arbitrary surface is mathematically a pseudo-scalar, its mechanical effect requires it physically to be a true scalar. Therefore, by contradiction this normal component must vanish, which in turn shows that the couple-stress tensor is skew-symmetric. Consequently, the pseudo couple-stress tensor has a true vectorial character.

Theoretical and Applied Mechanics, 2008

An objective of this paper is to reconcile the "symmetry" approach with the "symmetry groups" approach as these two different points of view presently coexist in the literature. Here we will be concerned exclusively with linearly elastic materials. The starting point for an analysis of the inherent symmetry of elastic materials is the notion of a symmetry transformation. Particularly, we paid attention to the compliance tensor for cubic and hexagonal crystals.

HAL (Le Centre pour la Communication Scientifique Directe), 2019

Tensor calculus is introduced to Physics and Mechanical engineering students in 2D and 3D and applied to anisotropic elasticity such as in condensed matter physics approaching the subject from the practical tool aspect point of view. It provides powerful mathematical techniques to tackle many aspects of Vector Calculus, Continuum mechanics, Solid State Physics, Electromagnetism...

Scientific Visualization: Interactions, Features, …

We present a visual approach for the exploration of stress tensor fields. Therefore, we introduce the idea of multiple linked views to tensor visualization. In contrast to common tensor visualization methods that only provide a single view to the tensor field, we pursue the idea of providing various perspectives onto the data in attribute and object space. Especially in the context of stress tensors, advanced tensor visualization methods have a young tradition. Thus, we propose a combination of visualization techniques domain experts are used to with statistical views of tensor attributes. The application of this concept to tensor fields was achieved by extending the notion of shape space. It provides an intuitive way of finding tensor invariants that represent relevant physical properties. Using brushing techniques, the user can select features in attribute space, which are mapped to displayable entities in a three-dimensional hybrid visualization in object space. Volume rendering serves as context, while glyphs encode the whole tensor information in focus regions. Tensorlines can be included to emphasize directionally coherent features in the tensor field. We show that the benefit of such a multi-perspective approach is manifold. Foremost, it provides easy access to the complexity of tensor data. Moreover, including wellknown analysis tools, such as Mohr diagrams, users can familiarize themselves gradually with novel visualization methods. Finally, by employing a focus-driven hybrid rendering, we significantly reduce clutter, which was a major problem of other three-dimensional tensor visualization methods.

Journal of Mathematical Physics, 2002