Fracture Mechanics of Electromagnetic Materials: Nonlinear by Xiaohong Chen

By Xiaohong Chen

Fracture Mechanics of Electromagnetic Materials presents a entire evaluation of fracture mechanics of conservative and dissipative fabrics, in addition to a basic formula of nonlinear box conception of fracture mechanics and a rigorous remedy of dynamic crack difficulties concerning coupled magnetic, electrical, thermal and mechanical box amounts.

Thorough emphasis is put on the actual interpretation of primary ideas, improvement of theoretical types and exploration in their functions to fracture characterization within the presence of magneto-electro-thermo-mechanical coupling and dissipative results. Mechanical, aeronautical, civil, biomedical, electric and digital engineers attracted to program of the rules of fracture mechanics to layout research and sturdiness evaluate of shrewdpermanent constructions and units will locate this ebook a useful source.

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Additional resources for Fracture Mechanics of Electromagnetic Materials: Nonlinear Field Theory and Applications

Example text

48) where β= π σ . 49) If σ σ y is small, Eq. 48) can be approximated as πK dp =  I 8  σ y 2   . 50) This expression can be compared with the estimation of the plastic zone size by the Irwin approach (1961): 1K rp = 2ry =  I π  σ y 2   . 51) Wells (1961, 1963) suggested that fracture in metals occurs when the crack-tip opening displacement (CTOD) reaches a critical value. The CTOD can be calculated from the elastic field (Goodier and Field, 1963) as  8σ y a   ln(sec β ) . 52) Generally speaking, if the extent of the cohesive zone is small enough compared to characteristic dimensions, regardless of the force-separation law, the J-integral, the energy release rate, the stress intensity factor, and Fundamentals of Fracture Mechanics 23 the crack-tip opening displacement are all equivalent fracture mechanics parameters under small-scale yielding conditions, that is, δt J = G = K I2 E ' = ∫ σ (δ )d δ .

Mai and Cotterell (1986) also showed the following equivalence: we = J c , β wp = (1 / 4)dJ R / da for double-edge notched tension (DENT) and deep center notched tension (DCNT) specimens, and β wp = (1 / 2)dJ R / da for deep single-edge notched tension (DSEN) specimens. 8 Configuration Force (Material Force) Method The notion of the Newtonian force is clarified by its role in describing the motion of a body. By contrast, the concepts of the energy-momentum tensor (also referred to as the Eshelby stress tensor) and the configuration force (also referred to as the material force) are introduced in the interpretation of the evolution of material microstructures such as defects (Eshelby, 1951, 1956, 1970).

55) where π = P / ρ is the polarization per unit mass and ρ is the mass density. 6 Poynting theorem The Poynting vector, which represents the flux of the electromagnetic energy, is denoted by S = E × H in the laboratory frame RG and by S = E × H in the co-moving frame RC . The Poynting theorem in RG gives the identity H⋅ ∂B ∂D + E⋅ = − je ⋅ E − ∇ ⋅ S . 56) With use of this identity, the electromagnetic power density can be expressed in a new form em w =− ∂ em u f − ∇ ⋅ [S − v( E ⋅ P)] . 57) The Poynting theorem in RC gives the identity * * H ⋅ B + E ⋅ D = − je ⋅ E − ∇ ⋅ S .