The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical...

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process...

calculated using the J-integral, i.e. G = J {\displaystyle G=J} , using J = ∫ Γ ( W n 1 − t i ∂ u i ∂ x 1 ) d Γ , {\displaystyle J=\int _{\Gamma }\left(Wn_{1}-t_{i}\...

the stress intensity factor K {\displaystyle K} and the elastic-plastic J-integral. For plane stress conditions, the CTOD can be written as: δ t = ( 8 σ...

elastic-plastic fracture mechanics can be used with parameters such as the J-integral or the crack tip opening displacement. The characterising parameter describes...

The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}...

Integral theory as developed by Ken Wilber is a synthetic metatheory aiming to unify a broad spectrum of Western theories and models and Eastern meditative...

as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It...

analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may...

the J-integral for plane stress and plane strain in Mode I are the same: J = μ π a 2 4 . {\displaystyle J={\frac {\mu \pi a^{2}}{4}}.} The J-integral can...

The INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is a retired space telescope for observing gamma rays of energies up to 8 MeV. It was launched...

In politics, integralism, integrationism or integrism (French: intégrisme) is an interpretation of Catholic social teaching that argues the principle...

The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces...

to connect the J-integral to the stress intensity factor because G = J = ∫ Γ ( W   d x 2 − t ⋅ ∂ u ∂ x 1   d s ) . {\displaystyle G=J=\int _{\Gamma...

leads to the generalised definition of the J-integral in equations below. For a dislocation pile-up, the J-integral is the summation of the Peach–Koehler configurational...

Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ʒwa]), Luzin integral or Perron...

central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators...

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that...

mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. The different sine integral definitions are Si...

In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define...

In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0...

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function...

mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations...

: 13–15  Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad...

The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related...

mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an...

In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...

mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students...

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied...

the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form...