Shear Deformation Terms Research Articles

Simple shear deformation and quartz crystallographic fabrics: a possible natural example from the Torridon area of NW Scotland

In this paper the microstructures and crystallographic fabrics of a 30 cm wide deformed quartz vein from the Torridon area of NW Scotland are described. The vein, which lies along a joint separating two blocks of Proterozoic gneiss, was deformed by crystal-plastic processes when the rigid joint blocks moved in opposite directions parallel to the vein margins. Mylonitic foliation within this quartzose ductile shear zone locally displays the classic orientation variation expected for heterogeneous simple shear deformation. In thin section the vein is a mylonite composed of dynamically recrystallized, elongate quartz grains aligned oblique to the mylonitic foliation. Fabrics within the central part of the vein have been examined by optical and X-ray texture goniometry methods, together with Orientation Distribution Function (ODF) analysis. The a-axis fabric consists of a dominant single point maximum, orientated within the XZ plane at 9° to the lineation (X); this point maximum occupies a pole position to the corresponding single girdle c-axis fabric. For individual positions on the c-axis fabric skeleton, the orientations of potential crystallographic slip systems (slip plane and direction) relative to shear zone co-ordinates have been calculated by ODF analysis. The calculation of relative resolved shear stresses (Schmid factors) for the potential slip systems, assuming a simple shear kinematic framework, reveals an elegantly simple relationship between individual grain c-axis orientation and corresponding orientations of potential slip systems. These results are readily interpreted in terms of bulk simple shear deformation.

Vibration and buckling of bimodulus laminated plates according to a higher-order plate theory

This paper presents the governing equations of rectangular simply supported single-layer and two-layer cross-ply plates of bimodulus materials related to the higher order shear deformation terms. Natural frequency parameters and buckling coefficients obtained by the iteration method are compared with published results for ordinary thick plates. The Mindlin plate theory result can be obtained by simplifying the governing equation of the higher-order plate theory formulation. The effects of higher-order shear deformation terms on neutral surface locations, buckling coefficients and natural frequencies are evaluated by observing the difference between the present higher-order theory results and the Mindlin plate theory results. Also, the bending stress is shown to affect significantly the buckling coefficient K cr and the non-dimensional frequency parameter Ω.

Vibration of a bimodulus thick plate according to a higher-order shear deformation theory

Equations of motion of a simply supported rectangular bimodulus thick plate in which the higher-order shear deformation terms are included are derived. The governing equations obtained using the average stress method are solved in exact form. The natural frequencies are compared with the previous results obtained for ordinary thick plates, and with the neutral surface locations and natural frequencies of bimodular plates. From those comparisons, the effects of higher-order shear deformation terms on the neutral surface locations and the natural frequencies can be observed. Also, the percentage error sense improvement by higher-order deformation terms can be reduced from 1.4 to 0.09%.

Free vibrations of a thick spinning annular disk with distributed masses at the outer edge

This paper deals with the free vibration problem of a spinning annular disk with uniformly distributed masses attached to its outer edge. Mindlin's transverse shear deformation and rotary inertia terms are included in the eigenvalue formulation. The natural frequencies of the disk are computed by the Rayleigh-Ritz technique for the combination of boundary conditions with constraints at the inner edge and free at the outer boundary. The numerical results illustrating the effect of changing thickness for different values of edge masses have been presented in a non-dimensional form.

Bending of a bimodulus laminated plate based on a higher-order shear deformation theory

This paper presents an exact flexural analysis of rectangular simply supported single-layer and two-layer cross-ply plates of bimodulus materials. The governing equations of a bimodulus plate based on a higherorder shear deformation theory are simplified from the composite plate. The present analysis of displacements in flexure is compared with Bert's results and Turvey's results which are based on Mindlin plate theory. The in-plane stress and bending stress are included in the present study. All the present numerical results are compared with the Mindlin plate theory (first-order plate theory) results. From those comparisons, the effects of higher-order shear deformation terms on the neutral surface locations and the flexure displacements can be observed.

Large amplitude vibration of a beam based on a higher-order deformation theory

A higher-order formulation of large amplitude vibrations of simply supported beams subjected to initial stress is presented. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress of the neutral axis of the beam. By neglecting the higher-order terms and introducing the shear correction factor into governing equations, these equations can be reduced to Timoshenko beam theory (present first-order beam theory). The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations in terms of an unknown function of time. The Runge-Kutta method is used to obtain the period. Numerical results show that if the amplitude ratio increases, significant errors in the first-order beam theory can be observed, and that higher-order shear deformation terms are necessary to solve this problem.

Determination of natural frequencies of a thick spinning annular disk using a numerical Rayleigh–Ritz’s trial function

Consideration is given in this paper to the problem of a thick, spinning, annular disk constrained at its inner radius. Mindlin’s transverse shear deformation and rotary inertia terms are included in the eigenvalue formulation. In the modified Rayleigh–Ritz method used here, the ‘‘admissible’’ or ‘‘trial functions’’ are developed numerically by an iterative scheme so that all the forced, as well as natural, boundary conditions are fully satisfied. The natural frequencies for different mode shapes of the disk have been computed and are presented in a nondimensional form as a function of rotating speed.

Vibration and stability of an initially stressed laminated plate based on a higher-order deformation theory

Equations of motion for a laminated plate in a general state of nonuniform initial stress where the higher-order shear deformation terms are included are derived. The equations are derived by using the average stress method. The natural frequencies and buckling loads of cross-ply laminated plates subjected to initial stress are investigated. The initial stress is taken to be a combination of uniaxial pure bending and extensional stresses in the plane of the plate. The results obtained from the present higher-order theory are compared with first-order theory and other higher-order theory results. In addition, the effects of various parameters on the natural frequencies and buckling loads are studied.

Finite element analysis for the transient response of thin elastic spherical shells

This paper presents the development of a finite element for the analysis of spherical shells. Twelve degrees of freedom are taken at each node of the element. The derivation of stiffness and consistent mass matrices is based on the improved shell theory which takes into account the effects of rotary inertia and shear deformation terms. Normal modes and the values of the nondimensional frequency parameter for the free axisymmetric vibration of the spherical shell are calculated using an iterative technique and the orthogonality conditions. These results are then compared with those obtained earlier from the closed form solution of the differential equations of motion. As evident, the present method is found extremely simplified compared to the exact solution of the frequency equation which is transcendental in nature and involves Lagendre functions of complex order. The maximum difference in the results is within 3%–4%. The forced motion of spherical shells subjected to impulsive loadings is also studied in this paper. Detailed numerical results are presented and discussed for various cases of loading conditions.

Validity of Thin-Plate Theory in Dynamic Viscoelasticity

The exact equations of dynamic viscoelasticity are used to determine the resonance damping for a vibrating plate, which is assumed as isotropic and homogeneous. To determine the validity of thin-plate theory, the same problem is investigated by means of the classical thin-plate equation that does not include shear deformation or rotatory inertia terms. The comparison between the thin-plate theory and the exact theory is carried out for two types of materials that correspond to real and complex values of the bulk modulus. In each case, the shear modulus is μ(1+ig), and values of g to 0.1 were used in the calculations. The two theories are in excellent agreement in a range of wavelengths as low as 10 times the thickness. It is found that thin-plate theory evaluates the damping more accurately than the static rigidity. [This work was performed at the Adelphi Research Center, Garden City, New York, under a contract with the Aeronautical Systems Division of Wright-Patterson Air Force Base, Ohio, AF 33(616)-8274, with M. A. Biot as Scientific Director.]

Axially Symmetric Flexural Vibrations of a Circular Disk

Abstract At high frequencies, the flexural vibrations of a plate are described very poorly by the classical (Lagrange) theory because of neglect of the influence of coupling with thickness-shear vibrations. The latter may be taken into account by inclusion of rotatory inertia and shear-deformation terms in the equations. The resulting frequency spectrum is given, in this paper, for the case of axially symmetric vibrations of a circular disk with free edges and is compared with the spectrum predicted by the classical theory.