Displacement Potential Approach Research Articles

Validating bond-based peridynamic model using displacement potential approach

Although peridynamics is widely used to investigate mechanical responses in materials, the ability of peridynamics to capture the main features of realistic stress states remains unknown. Here, we present a procedure that combines analytic investigation and numerical simulation to capture the elastic field in the mixed boundary condition. By using the displacement potential function, the mixed boundary condition elasticity problem is reduced to a single partial differential equation which can be analytically solved through Fourier analysis. To validate the peridynamic model, we conduct a numerical uniaxial tensile test using peridynamics, which is further compared with the analytic solution through a convergence study. We find that, when the parameters are carefully calibrated, the numerical predicted stress distribution agrees very well with the one obtained from the theoretical calculation.

Thermo-Chemo-Elasticity Considering Solid State Reaction and the Displacement Potential Approach to Quasi-Static Chemo-Mechanical Problems

Engineering materials and structures represent complex behaviors when reacting to superposed influences of mechanical forces, high temperature, diffusion and reaction of chemicals, which could cause large internal stresses and further induce cracks or failure. To determine the material reliability and integrity, the multi-field interactions and stresses/strains evolutions need to be identified at first. We proposed a theory of thermo-chemo-elasticity considering solid state reactions between the solid phase and absorbed chemicals in a stressed-solid. Both diffusion–reaction induced chemical strains and thermal dilations are taken into account as functions of species concentration, reaction extent and temperature. The fully coupled conservation laws, constitutive equations and chemical kinetics are formulated for the initial-boundary problem. For isotropic solids, we developed a displacement potential approach for steady-state 3D problems of thermo-chemo-elasticity. Solutions can be found from particular solutions of displacement potential and homogeneous solution of thermo-chemo Lamé equation. This approach is also available for transient chemo-mechanical problems in thermal equilibrium providing that quasi-static conditions are introduced. We exemplified the model with a reaction-dominated problem of a core–shell structure subjected to chemo-mechanical loading and the results demonstrate the capability of the model in dealing with comprehensive influences of solid state reaction and species diffusion on solids.

Analytical Solution of Mixed Boundary Value Problems Using the Displacement Potential Approach for the Case of Plane Stress and Plane Strain Conditions

Two elastic plate problems made of duralumin are solved analytically using the displacement potential approach for the case of plane strain and plane stress conditions. Firstly, a one end fixed plate is considered in which the rest of the edges are stiffened and a uniform load is applied to the opposite end of the fixed end. Secondly, a plate is considered in which all of the edges are stiffened and a uniform tension is applied at its both ends. Solutions to both of the problems are presented for the case of plane stress and plane strain conditions. The effects of plane stress and plane strain conditions on the solutions are explained. In the case of stiffening of the edges of the plate, the shape of the plate does not change abruptly, which is clearly observed in both of the cases. For the plane strain condition, the plates become stiffer in the loading direction as compared to the plane stress condition. For the plane strain condition, there is a significant variation of the normal stress component, σ<sub>zz</sub> at different sections of the plate. The graphical results, clearly identify the critical regions of the plate for the case of the plane stress and plane strain condition.

An alternative numerical solution of 900/00/900 cross-ply laminated composite plate using displacement potential approach

ABSTRACTConventional loading with a composite plate has a very small moment of inertia due to its low height, and it was solved by shear deformation and other approaches. In the present study, a thin cross-ply 900/00/900 laminated composite plate is erected to increase its height, which increases its moment of inertia with load; increased moments of inertia of the plate increase its load-carrying capacity in a real structure. Finite difference solutions of a cross-ply 900/00/900 laminated plate made of T-300 carbon/epoxy are obtained using displacement potential approach. The plate behaves like a cantilever beam; one end of the plate is rigidly fixed and a uniformly distributed load is applied on its top span. The displacement potential approach is extended using the lamination theory of composite materials and plane stress concept. The different equivalent displacement, strain, and stress components at different sections of the plate with its deformed shape are discussed. Using the concept of the classical theory of laminations, different stress components along the lamina and inter-laminar regions are obtained. Besides to verify the reliability and soundness of the present numerical approach, the solutions of the present problem are compared with those of the finite element method.

Influence of surface energy on the elastic compression of nanosphere

The influence of surface energy on the elastic compression of nanosphere is addressed through the principle of minimum potential energy. By using the displacement potential approach, the elastic field of nanosphere under diametrical compression is derived analytically. Firstly, surface energy induces a uniform pre-existing hydrostatic compression in the entire nanosphere. More importantly, when the ratio of surface energy density to the radius of sphere is comparable with the elastic modulus, the response of nanosphere to external loading will be evidently distinct from the classical prediction. When the compressive load-depth curve is used to calculate the elastic modulus of nanosphere, the presence of surface energy predicts the size dependence of elastic modulus, which enhances significantly as the radius of sphere decreases below 100 nm. This study provides an efficient tool to analyze the elastic deformation of nanoparticles and measure their elastic properties.

A Numerical Solution of a One End Fixed Glass/Epoxy Plate Having a Circular Cutout Subjected to a Uniform Shear Using Displacement Potential Approach

The finite-difference technique based on the displacement potential approach of orthotropic composite materials is extended to solve elastic plane stress problems of orthotropic composite materials with geometric perturbations, such as holes, arbitrary defects, notches, etc. In this analysis, one fixed elastic glass/epoxy plate having an internal hole is considered and a uniform shear load is applied to the opposite end of the supporting edge. Critical sections of the plate are identified with the detailed discussions of the elastic field of the plate. Effects of sizes of the holes of the plate on the elastic field are also discussed with the help of graphical solutions. The reliability of the extended finite-difference technique based on the displacement potential approach of orthotropic composite materials is shown by the comparison of solutions between the FDM and FEM.

Effects of Fiber Orientation and Material Isotropy on the Analytical Elastic Solution of a Stiffened Orthotropic Panel Subjected to a Combined Loading

Using the displacement potential approach of orthotropic composite materials for the plane stress conditions, an orthotropic panel subjected to a combined loading at its right lateral edge is solved. Effects of fiber orientations and material isotropy on the elastic field of an orthotropic panel subjected to a combined loading are discussed. The analytical elastic solutions at different sections of the panel with fiber orientationθ=90° are compared with those of finite element predictions to ensure the reliability of our present solutions.

A finite-difference solution of boron/epoxy composite plate with an internal hole subjected to uniform tension/displacements using displacement potential approach

Finite-difference technique is extended to obtain the stress as well as the displacement components with the deformed shape of a boron/epoxy orthotropic composite plate with an internal hole subjected to uniform tension (case 1)/displacements (case 2) at its two opposite ends. Critical regions are identified by the detailed discussions of the elastic field of the plate problem for both of the cases. Effects of fiber orientation and sizes of the hole on the elastic field of the plate for case 2 are discussed. Boundary condition effects on the elastic field of the plate for case 2 are also investigated. To check the reliability, appropriateness, and soundness of the present approach, firstly the tangential stress component, σϕ along the circular cavity of the plate for case 1 obtained by the finite difference method is compared with that of the classical theory of anisotropic elasticity and secondly the different stress components at the section, y/b=0 and 0.5 of the plate of orthotropic composite materials for case 2 obtained by the finite difference method based on displacement potential approach are compared with those of Finite element predictions by the commercially available ANSYS Finite element software.

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.

Analytical solution of a stiffened orthotropic plate using alternative displacement potential approach

Analytical solution of elastic fields of a stiffened composite plate subjected to axial tension and pure bending is obtained using alternative displacement potential approach. In this approach, the problem is formulated in terms of a single potential function, defined in terms of the displacement components, which satisfies a single differential equation of equilibrium. All the parameters of the elastic field are expressed into algebraic equations using Fourier series. Assuming a trigonometric function as the displacement potential solution to the present problem, the mixed-boundary conditions at the stiffened edges are automatically satisfied. The resulting states of stress and displacement in the composite plate due to axial tension and pure bending are discussed in details. The effect of fibre orientation on the elastic field is also investigated.

A finite difference solution of a steel plate with an internal hole using the displacement potential approach

The finite difference technique based on a displacement potential formulation is extended to solve an elastic plane stress problem with geometric perturbations such as holes, arbitrary defects, notches, etc. An elastic steel plate having an internal hole is constrained at one edge and a shear load is applied to the opposite edge. The present problem is solved by using the extended finite difference technique based on displacement potential approach. The different displacement and stress components at different sections of the plate are shown graphically and discussed; from the analysis of stress and displacement components at different sections of the plate, critical regions of the plate are identified. The superiority of the finite difference method (FDM) based on the displacement potential formulation over the finite element method (FEM) is verified by the comparison between the solutions of the present problem by the FDM and FEM. From the comparison between the solutions of the FEM and FDM, it is concluded that the present finite difference technique based on the displacement potential formulation is reliable and can be used to solve the structural elements having internal defects.

Analysis of the Effect of Fiber Orientation on the Elastic Field in a Stiffened Orthotropic Panel under Uniform Tension using Displacement Potential Approach

The effect of fiber orientation on the stresses and displacements at different sections of a stiffened panel of orthotropic composite material under a uniform tension is analyzed. The supporting edge of the panel is rigidly fixed and two opposing edges are stiffened. The mixed boundary value plane stress elasticity problem is formulated in terms of a single displacement potential function ψ, defined in terms of the two displacement components. The solution of the problem is obtained in terms of the Fourier infinite series for two different fiber orientations of the orthotropic panel. Analysis of the results shows that the fiber orientation has significant effect on the distribution of stress and displacement components at different sections of the panel.

Analytical solution to a mixed boundary value elastic problem of a roller-guided panel of laminated composite

This study presents an analytical solution to elastic field in a roller-guided panel of symmetric cross-ply laminated composite material. The mixed boundary value two-dimensional plane stress elasticity problem is formulated in terms of a single displacement potential function. This reduces the problem to the solution of a single fourth order partial differential equation of equilibrium as the other equilibrium equation is satisfied automatically. The solution is obtained in terms of an infinite Fourier series. To present some numerical results, a panel of glass/epoxy laminated composite is considered and different components of stress and displacement at different sections of the panel are presented graphically. To justify the present analytical solution, it is compared with the finite element solution obtained by using the commercial software ANSYS. It is found that the two solutions agree well with each other. This ensures that the formulation developed in this study based on the displacement potential approach can be used to obtain analytical solution of an elastic field in structural elements of laminated composite under any mode of boundary conditions prescribed in terms of either stress, displacement or any combination of these.

Displacement potential solution of a deep stiffened cantilever beam of orthotropic composite material

An analytical solution of the elastic field of a deep stiffened cantilever beam of orthotropic composite material is presented in the paper. The cantilever beam is subjected to a parabolic shear loading at its free lateral end and the two opposing longitudinal edges are stiffened. Unidirectional fibre-reinforced composite is considered for the present analysis where the fibres are assumed to be directed along the beam length. Following a new development, the present mixed-boundary-value elastic problem is formulated in terms of a single potential function defined in terms of the associated displacement components. This formulation reduces the problem to the solution of a single fourth-order partial differential equation of equilibrium and is capable of dealing with mixed modes of boundary conditions appropriately. The solution is obtained in the form of an infinite series. Results of different stress and displacement components at different sections of the composite beam are presented numerically in the form of graphs. Finally, in an attempt to check the reliability as well as the accuracy of the present solution, the problem is solved by using two standard numerical methods of solution. A comparison of the results shows that the analytical and numerical solutions of the present problem are in good agreement and thus establishes the soundness as well as the reliability of the present displacement potential approach to solution of the elastic field of orthotropic composite structures.

Displacement potential approach to the solution of stiffened orthotropic composite panels under uniaxial tensile load

The current paper presents an analytical method for the analysis of elastic field in a stiffened unidirectional orthotropic composite panel under uniaxial tensile load in the direction of fibres. The mixed boundary value elasticity problem is formulated in terms of a single potential function ψ, expressed in terms of the displacement components. This function satisfies one of the differential equations of equilibrium automatically. Therefore, the mixed boundary value problem is reduced to the solution of a single differential equation of equilibrium along with the associated boundary conditions. The analytical solution is obtained in the form of an infinite series. Some numerical results of different stress and displacement components at different sections of the panel are presented in the form of graphs. The reliability and soundness of the method are verified by comparing the results with finite-difference and finite-element solutions of the problem. Good agreement of the results establishes the superiority of the present analytical method based on the displacement potential function.

Mixed Mode Dynamic Fracture in Particulate Reinforced Functionally Graded Materials

A detailed analytical and experimental investigation is presented to understand the dynamic fracture behavior of functionally graded materials (FGMs) under mode I and mixed mode loading conditions. Crack-tip stress, strain and displacement fields for a mixed mode crack propagating at an angle from the direction of property gradation were obtained through an asymptotic analysis coupled with a displacement potential approach. This was followed by a comprehensive series of experiments to gain further insight into the behavior of propagating cracks in FGMs. Dynamic photoelasticity coupled with high-speed photography was used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the FGMs. Dynamic fracture experiments were performed using different specimen geometries to develop a dynamic constitutive fracture relationship between the mode I dynamic stress intensity factor (K ID ) and crack-tip velocity ( $${\mathop a\limits^ \cdot }$$ ) for FGMs with the crack moving in the direction of increasing fracture toughness. A similar $${\mathop a\limits^ \cdot }$$ -K ID relation was also obtained for matrix material (polyester) for comparison purposes. The results obtained show that crack propagation velocities in FGMs were about 80% higher than the polyester matrix. Crack arrest toughness was found to be about 10% lower than the value of local fracture toughness in FGMs.

Displacements, strains and stresses associated with propagating cracks in materials with continuously varying properties

Crack tip stress, strain and displacement fields for a propagating crack along the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with a displacement potential approach. The analysis for the opening mode is performed assuming two types of property variation: (a) linearly varying shear modulus with constant density, and (b) exponentially varying shear modulus and density. The first six terms in the series expansion of the stress, strain and displacement fields for the dynamic crack are derived to explicitly bring out the influence of nonhomogeneity on the structure of the displacement, strain and stress fields. The analysis revealed that crack tip stress fields retain the inverse square root singularity and only the higher order terms in the expansion are influenced by material inhomogeneity. Using these stress, strain and displacement fields, contours of constant maximum shear stress, constant first stress invariant and constant in-plane displacements are generated and the effect of the nonhomogeneity parameter on these contours is discussed.

Dynamic damage growth in particle reinforced graded materials

A scheme for preparing materials with continuous-graded properties using polyester resin and cenospheres is presented. The elastic and fracture property profiles of these materials are characterized and the results are discussed. Crack-tip stress, strain and displacement fields for a transient crack propagating along the direction of property gradation in functionally graded materials are obtained through an asymptotic analysis coupled with a displacement potential approach. The influence of the transient nature of crack-tip on contours of constant maximum shear stress, constant first stress invariant and constant in-plane displacements is discussed. Results of initial experimental investigations carried out using dynamic photoelasticity in conjunction with high-speed photography to characterize the behavior of a propagating crack in functionally graded materials are presented.

Dynamic damage growth in particle reinforced graded materials

A scheme for preparing materials with continuous-graded properties using polyester resin and cenospheres is presented. The elastic and fracture property profiles of these materials are characterized and the results are discussed. Crack-tip stress, strain and displacement fields for a transient crack propagating along the direction of property gradation in functionally graded materials are obtained through an asymptotic analysis coupled with a displacement potential approach. The influence of the transient nature of crack-tip on contours of constant maximum shear stress, constant first stress invariant and constant in-plane displacements is discussed. Results of initial experimental investigations carried out using dynamic photoelasticity in conjunction with high-speed photography to characterize the behavior of a propagating crack in functionally graded materials are presented.

Numerical evaluation of three-dimensional water impact by the displacement potential formulation

The numerical accuracy of the displacement potential approach for a three-dimensional water-impact problem is investigated. An examination of the elliptic paraboloid shows that the boundary-value problem for the displacement potential, as well as the velocity potential, can be solved accurately, even if the panel size of the numerical scheme is no small enough. The numerical accuracy of the position of the contact line is poor. However, the accuracy of the virtual mass is good because of the averaging effect along the contact line. A comparison with Scolan and Korobkin's designed body confirms the accuracy of the drop speed and the penetration displacement. The displacement potential approach is extended to the case of water impact with trapped air. The numerical accuracy of the calculated air pressure in the cavity is confirmed by comparison with an experimental result. With these results, the suitability of the displacement potential approach to the ship-slamming problem is demonstrated.