China
Africa
Explore chapters and articles related to this topic
Background Discussion
Published in George Emanuel, Analytical Fluid Dynamics, 2017
where the volume V is fully enclosed by S, n^ $ \hat{n} $ is an outward unit normal vector to S, ϕ(r→) $ \phi (\vec{r}) $ is an arbitrary scalar function, and A→(r) $ \vec{A}(r) $ is an arbitrary vector function. This result is actually three separate equations combined into a single convenient form. Of these three equations, we shall make explicit use of the first two. Note that the surface integral is a double integral, while the volume integral is a triple integral. We shall also need a dyadic version of the middle equation, given by ∫V∇·Φ↔dv=∮Sn^·Φ↔ds $$ \int\limits_V \nabla \cdot {\mathop{\Phi}\limits^{\leftrightarrow} } dv = \oint\limits_S \hat n\cdot {\mathop{\Phi}\limits^{\leftrightarrow} } ds $$
Interpretation of saddle-splay and the Oseen-Frank free energy in liquid crystals
Published in Liquid Crystals Reviews, 2018
One of the oldest problems in liquid-crystal research is to characterize how the director field can be distorted away from a uniform state, or in other words, to identify the elastic modes of a nematic liquid crystal. Through the mid-twentieth century, this issue was investigated in classic work by Oseen [1] and Frank [2], and further by Nehring and Saupe [3,4]. This body of research led to the Oseen-Frank free energy density, which is discussed in many textbooks, such as Ref. [5], and which is widely used in liquid-crystal science and technology. This free energy density includes terms representing the cost of three distortion modes – splay, twist, and bend – and also includes a further term called saddle-splay. The saddle-splay contribution to the free energy density is the total divergence of a vector field. As a result, the volume integral of this term can be transformed into a surface integral. For that reason, the saddle-splay contribution is often considered as surface elasticity, in contrast with the splay, twist, and bend contributions, which are bulk elasticity.
Automatic differentiation of a finite-volume-based transient heat conduction code for sensitivity analysis
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Equation (11) is discretized using the finite-volume method by integrating the equation over a discrete polyhedral control volume in space, denoted ΩP. Let ΩP have volume VP and be bounded by the control surface which is the union of the discrete control surfaces . Each control surface has area Aip, where and Nip is the number of discrete control surfaces. Carrying out this procedure on Eq. (11) results in where divergence theorem has been used to convert the volume integral of the term on the right into a surface integral. The discretization given in Eq. (12) can be shown to be second-order accurate in space and time, provided the points P and ip are located at the centroids of the control volume and control surfaces, respectively, and all further interpolations are made with second-order accuracy.
Reliability study on fracture and fatigue behavior of pavement materials using SCB specimen
Published in International Journal of Pavement Engineering, 2020
Muhammad Mubaraki, H. E. M. Sallam
In the present work, the domain integral method is commonly used to extract stress intensity factors (SIFs). In a finite element model, SIF can be thought of as the virtual motion of a block of material surrounding each node along the crack line. Each such block is defined by contours: each contour is a ring of elements completely surrounding the nodes along the crack line from one crack face to the opposite crack face. These rings of elements are defined recursively to surround all previous contours. ABAQUS/ Standard automatically finds the elements that form each ring from the regions given as the crack-line definition. Each contour provides an evaluation of the contour integral. Using the divergence theorem, the contour integral can be expanded into a volume integral, over a finite domain surrounding the crack. This domain integral method is used to evaluate contour integrals in ABAQUS/ Standard. The method is quite robust in the sense that accurate contour integral estimates are usually obtained even with quite coarse meshes; because the integral is taken over a domain of elements surrounding the crack, errors in local solution parameters have less effect on the evaluated quantities such as the stress intensity factors. The stress intensity factors KI, KII and KIII (mode I, mode II, and mode III SIF) are usually used in linear elastic fracture mechanics to characterise the local crack-tip/crack-line stress and displacement fields. They are related to the energy release rate (the J-integral). The energy release rate is calculated directly in ABAQUS/ Standard.