Numerical Evaluation of Integrals

Fuller, William R.

doi:10.1007/978-1-4612-9938-7_10

Cited by 7 publications

(1 citation statement)

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“…............................. (5) It was assumed that n may be determined from the quotient of the actual Young modulus (E) and hypothetical Young modulus (E max ). As there is a relation between the velocity of ultrasonic wave and Young modulus (if there is no change in density) the values of n were determined from the expression: 6where: c L is longitudinal ultrasonic wave velocity, c L max is maximum ultrasonic wave velocity.…”

Section: Concept Of Utilizing the Elongation Curve For Brittle Materialsmentioning

confidence: 99%

Quality Control by Means of Ultrasonic in the Production of Ductile Iron

2010

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The paper presents several concepts of physical substantiation for the rationale of building the relationships between mechanical properties and ultrasonic inspection indicators. Special emphasis is put on assessing the influence of graphite shape number S S and the number of graphite precipitations N A on the tensile strength and longitudinal ultrasound wave velocity of ductile iron manufactured under production conditions. Tests were conducted on wedge casts which were used as samples for tensile tests and a map of the structure and longitudinal ultrasound wave velocity was determined for the cast wedges. The tensile tests were conducted and values of graphite shape number, average number of graphite precipitations and longitudinal wave velocity were determined in the place where the sample was broken. Relations between the mechanical properties, graphite shape number, number of graphite precipitations and the velocity of longitudinal ultrasonic wave were determined.KEY WORDS: ductile iron; ultrasonic method; shape and number of graphite precipitations; mechanical properties.tween tensile strength and the velocity of ultrasonic wave was also undertaken in the paper. 9) On the basis of the elongation curve of a brittle material, the author assumed that: where: UTS min , minimum tensile strength corresponding to the elastic strain range; n, fraction of tensile strength corresponding to the plastic strain range (Fig. 1).Rearranging this assumed solution, we get: It was assumed that n may be determined from the quotient of the actual Young modulus (E) and hypothetical Young modulus (E max ). As there is a relation between the velocity of ultrasonic wave and Young modulus (if there is no change in density) the values of n were determined from the expression:........................ (6) where: c L is longitudinal ultrasonic wave velocity, c L max is maximum ultrasonic wave velocity.On substitution of formula (6) into (5), we get a relationship:The values A and B are characteristic for cast iron. Despite rather controversial assumptions in the paper 6) the obtained relationship was proven to be useful for the assessment of the tensile strength of grey cast iron. Concept of Utilizing the Law of MixturesFor their description of the relationship between material structure and its property, authors of many papers employed the co-called "law of mixtures", defined as:where: W, property of material; W a , W b , properties of aand b-phases, respectively; V Va , V Vb , volume fractions of a-and b-phases, respectively.The law is derived from basic relations in stereology. Those relationships can be formulated as follows: a fraction of the length of unitary section corresponding to flat crosssections of grains of a tested phase (l a /l), fraction of surface occupied by this phase on the plane occupied by this phase on a unitary area of polished microsection (a a /a), and a fraction of volume occupied by a specific phase in the unit of alloy volume (V a /V) are expressed by the same quantity.The law of mixtures is most often use...

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Section: Concept Of Utilizing the Elongation Curve For Brittle Materialsmentioning

confidence: 99%

Quality Control by Means of Ultrasonic in the Production of Ductile Iron

2010

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A Moment Matrix Approach to Multivariable Cubature

Fialkow

Petrović

2005

Integr. equ. oper. theory

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Abstract. We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure µ, the bound is based on estimating ρ(C) := inf{rank (T −C) : T Toeplitz and T ≥ C}, where C := C [µ] is a positive matrix naturally associated with the moments of µ. We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag (c1, . . . , cn) (including the case when µ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps c k > c k+1 .

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