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In this article, an analytical method is developed to obtain the response of magnetothermoelastic stress and perturbation of the magnetic field vector for a thick-walled spherical functionally graded materials (FGM) vessel. The vessel, which is placed in a uniform magnetic field, is subjected to an internal pressure and transient temperature gradient. Using the Hankel and Laplace transform techniques, the dynamic equation of magnetothermoelastic is solved and the radial and circumferential stresses as well as the perturbation of the magnetic field vector for a typical material are obtained. Moreover, the effect of magnetic field vector and material inhomogeneity on the stresses is investigated.Keywords Magnetothermoelasticity · Functionally graded material · Thick hollow sphere · Perturbation of magnetic field vector · Transient response List of symbols U , u r displacement vector and radial displacement (m), σ rr , σ θθ radial and circumferential stresses (N/m 2 ) λ 0 , G 0 , γ 0 , ρ 0 , and Lame's constant (N/m 2 ), shear modulus (N/m 2 ), thermal µ 0 modulus (N/Km 2 ), mass density (kg/m 3 ) and magnetic permeability coefficient (H/m), respectively m material grading exponent r radius (m) T (r, t) temperature change (K) t, a, b time (s), inner and outer radii of the FGM hollow sphere (m) H , h magnetic intensity vector, perturbation of magnetic field vector J electric current density vector e perturbation of electric field vector f φφ Lorentz's force (kg/m 2 , s 2 ) P i (t), P o (t) internal and external pressure of FGM hollow sphere (N/m 2 ) u r 0 (r ), v r 0 (r ) initial radial displacement (m), initial speed (m/s) V e elastic wave speed of homogeneous sphere (m/s) V t magnetothermoelastic wave speed (m/s) ω natural frequency of the FGM hollow sphere (1/s)

The main aim of this article is to propose some analytical and numerical expressions for evaluation of the critical value of the J-integral (Jcr) in plates weakened by blunt V-notches under Mode I loading for brittle or quasi-brittle materials. The critical J-integral is a material property in cracks. In notches, however, the specimen geometry also affects this parameter. In analytical expression, the relationship between J-integral and strain-energy density has been applied in order to evaluate the Jcr. The strain-energy density distribution over the semicircular arc of the notch (i.e. the parameter δ proposed in some previous articles) has also been used to derive a numerical equation for evaluation of the Jcr. The results have shown that the normalized Jcr (Jcr/JIC), that is, the ratio of the critical J-integral in notches to one in cracks, is a function of RC/ρ ratio, notch angle (2α), and Poisson’s ratio. The loading condition (bending or tensile loading), the ratio of the specimen width to the notch depth (w/a), and the notch acuity (a/ρ) have negligible effect on the normalized Jcr, for constant values of other parameters. Good agreement was found in critical fracture load evaluated by means of the Jcr criterion (using the proposed equations) with experimental data taken from the literature for blunt V-notches under Mode I loading.