Dynamic Elastic Solutions Research Articles

Nonlinear dynamic response of sandwich wide panels

This work focuses on the general nonlinear dynamic formulation of sandwich panels based on the Extended High-order Sandwich Panel Theory. The faces and the core are both considered undergoing large deformation with moderate rotation. In the literature, when it comes to nonlinear dynamic response, various simplifying assumptions are adopted in the nonlinear kinematic relations. A critical assessment of these simplifications, as well as a comprehensive investigation of the geometric nonlinearity effects on the dynamic response of sandwich beams/wide panels is presented. The Extended High-order Sandwich Panel Theory (EHSAPT)-based finite element is used to formulate the equations of motion, and the time response is obtained by the central difference method. The transient response of a sandwich panel subjected to a distributed blast loading is studied as a numerical example. By neglecting the nonlinear terms, the nonlinear dynamic analysis is reduced to the linear dynamic small deformation analysis, for which a closed-form elasticity solution exists. This, the linear response is first compared with the closed-form linear dynamic elasticity solution. It is shown that the linear EHSAPT yields an identical response to the dynamic elasticity analysis. Subsequently, the dynamic response is evaluated by considering the geometric nonlinearities. Several simplified nonlinear models with partial nonlinear terms included are considered. These models reflect the assumptions commonly adopted in the literature. Numerical results are presented to investigate the effects of geometric nonlinearities on the displacements and stresses. It is shown that the geometric nonlinearities have a significant effect on the dynamic response of sandwich panels and, in particular, the nonlinear terms in the core kinematic equations are especially important.

Dynamic analysis of a laminated cylindrical shell with piezoelectric layer and clamped boundary condition

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to exponential dynamic load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity partial differential equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in axial direction. The resulting equations are reduced to a system of ODE with constant coefficients by means of Galerkin׳s integral method and finally the governing equations are solved by using linear finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The ratio effect of the radius to thickness on dynamic behaviour is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented.

Elasticity Solution of a Laminated Clamped Cylindrical Shell with Piezoelectric Layer under Dynamic Load

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to impulse load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity PDE equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in the axial direction. The resulting equations are transferred into state space form and reduced to an eigenvalue problem by using Galerkin's finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The radius to thickness ratio effect on dynamic behavior is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented and natural frequencies are compared with simply-supported shell results.

Transient Blast Response of Sandwich Plates by Dynamic Elasticity

The three-dimensional linear dynamic elasticity problem formulation and solution for a generally asymmetric sandwich plate consisting of core and face sheets that are orthotropic, and subjected to blast loading, is presented. Laplace transforms are used to obtain ordinary differential equations in the complex Laplace space (with the variable being the through-thickness coordinate), which are subsequently solved in closed form for a simply supported plate, with the solution involving a cubic characteristic equation with complex coefficients. Subsequently, the time response is obtained by a numerical inverse Laplace transform by use of the Euler method. A realistic material and blast case is used to demonstrate the transient behavior for the displacements and face sheet/core interfacial transverse normal and shear stresses. The elasticity results are compared with the predictions of the first-order shear deformation plate theory as well as a high-order sandwich panel theory. This dynamic elasticity solution can serve as a benchmark in assessing the accuracy of sandwich-plate theories.

Blast Response of a Sandwich Beam/Wide Plate Based on the Extended High-Order Sandwich Panel Theory and Comparison With Elasticity

This paper presents a one-dimensional analysis for the blast response of a sandwich beam/wide plate with a compressible core. The dynamic version of the recently developed extended high-order sandwich panel theory (EHSAPT) is first formulated. Material, geometric, and loading parameters are taken from blast experiments reported in literature. The novelty of EHSAPT is that it includes axial rigidity of the core and allows for three generalized coordinates in the core (the axial and transverse displacements at the centroid of the core and the rotation at the centroid of the core) instead of just one (shear stress in the core) of the earlier high-order sandwich panel theory (HSAPT). The solution procedure to determine the dynamic response to a general load applied on the top face sheet of a general asymmetric simply supported configuration is outlined. Although the dynamic EHSAPT is formulated in its full nonlinear version, the solution is for the linear problem so the accuracy of EHSAPT, along with the other theories, can be assessed by comparison to an available dynamic elasticity solution. Results show that the EHSAPT is very accurate and can capture the complex dynamic phenomena observed during the initial, transient phase of blast loading.

Dynamic Elasticity Solution for the Transient Blast Response of Sandwich Beams/Wide Plates

The linear dynamic elasticity problem formulation and solution for a generally asymmetric sandwich beam/wide plate consisting of orthotropic core and face sheets subjected to blast loading is presented. The Laplace transform is used to obtain ordinary differential equations in the complex Laplace space (the variable being the through-thickness coordinate), which are subsequently solved in closed form for a simply supported beam. The time response is then obtained by a numerical inverse Laplace transform by using the Euler method. The results for realistic material and blast cases for the transient displacements and face sheet/core interfacial transverse normal and shear stresses are presented. A comparison of the elasticity results is also made with first-order shear and high-order sandwich panel theories. This solution can be used as a benchmark in assessing the accuracy of advanced sandwich panel theories.

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis.

Investigation of the scattering of harmonic elastic waves by two collinear symmetric cracks using non-local theory

In this paper, the scattering of harmonic waves by two collinear symmetric cracks is studied by use of non-local theory. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic problem to obtain the stress occurring at the crack tips. The Fourier transform is applied and a mixed boundary-value problem is formulated. The solutions are obtained by means of the Schmidt method. This method is more exact and more appropriate than Eringen's for solving this kind of problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the circular frequency of the incident wave.

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.

Investigation of the scattering of harmonic elastic waves by two collinear symmetric cracks using the non-local theory

The scattering of harmonic waves by two collinear symmetric cracks is studied using the non-local theory. A one-dimensional non-local kernel was used to replace a twodimensional one for the dynamic problem to obtain the stress occurring at the crack tips. The Fourier transform was applied and a mixed boundary value problem was formulated. Then a set of triple integral equations was solved by using Schmidt's method. This method is more exact and more reasonable than Eringe's for solving this problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the circular frequency of incident wave.

Investigation of the scattering of harmonic elastic shear waves by two collinear symmetric cracks using the non-local theory

In this paper, the scattering of harmonic shear waves by two collinear symmetric cracks is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then, a set of triple integral equations is solved using a new method, namely Schmidt's method. This method is simple and convenient for solving this problem. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length.

Investigation of the scattering of harmonic elastic shear waves by a finite crack using the non-local theory

In this paper, the scattering of harmonic shear waves by a finite crack is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt's method instead of using the first or the second Fredholm integral equation method. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack size.

Investigation of the scattering of harmonic elastic antiplane shear waves by a finite crack using the non-local theory

In this paper, the scattering of harmonic antiplane shear waves by a finite crack is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. Contrary to the classical elasticity solution, it is found that no stress singularity is presented at the crack tip. The non- local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length.

Dynamic elastic solutions in neo-Hookean and Mooney-Rivlin materials

Inhomogeneous motions are discussed within the context of non-linear constitutive theories. Deformations set up by time-dependent applied forces or displacements are analyzed for an infinite slab of neo-Hookean material and a circular cylinder of a Mooney-Rivlin type. The problems reduce to obtaining the solutions for simple linear differential equations.

The configuration of moving crack tip zones

The dynamic elasticity solution of a steady state crack is used for determination of the geometrical characteristics and of the displacement rates within the discrete crack zones which are formed as a result of the selective propagation of cleavage microcracks ahead of the tip of a running brittle crack in a mild steel plate. The zone length and the stress distribution in the zone are found to be strongly dependent on the assumed form of the stress-displacement relation for the progressively fracturing metal. On the other hand, the crack opening displacement rate is much less sensitive to the assumed form. The magnitudes of the nominal plastic strain rates, found by an approximate procedure, are of the order of 10 5 to 10 6 sec −1. A comparison with the results of dynamic tension tests on similar steels indicates that the flow stress at these strain rates would exceed the twinning stress, and therefore, it is suggested in agreement with experimental observations, that twinning is the principal deformation mode in the crack zone. Accordingly, the crack tip boundary conditions considered in the solution of crack propagation problems can be assumed as independent of crack velocity. Also, their possible dependence on temperature would not be related to the deformation mode in the crack zone.

Formation of plastic enclaves at running brittle cracks

Onset of enclave formation and their growth at the tip of a steady state crack which propagates under plane strain conditions in a mild steel plate is investigated. It is recognized that the crack tip boundary conditions are determined by the stresses transmitted in the weakened ductile links within a discrete crack zone which forms as a result of selective microcrack propagation in the elastic material ahead of the main crack. A general condition for the beginning of nonuniform viscoplastic yielding is proposed for an arbitrary path in the stress-time space and used for investigation of initial yielding and for an approximate description of enclave growth, while the loading path is derived from a dynamic elasticity solution of the crack problem. The plastic enclaves form initially at some distance from the crack line and may adjoin the crack zone only in later stages of their development. It is suggested that enclave formation and their eventual interference with the local separation process are respectively responsible for energy transition and crack arrest in brittle fracture.