Modified Contact Law Research Articles

Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body.

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

A variational iteration solution for elastic–plastic impact of polymer/clay nanocomposite plates with or without global lateral deflection, employing an enhanced contact law

In the present paper, characteristics of low/medium velocity impact responses of thin nanocomposite plates impacted by rigid spherical indenters are investigated. The plates may either have lateral deflections or resting on rigid substrates. It is obvious that for thin plates with transverse mobility subjected to impact with rigid indenters, the actual indentation value is much less than that predicted by the traditional Hertz-type laws. A modified contact law that takes into account effects of the plate thickness and boundary conditions is employed in the present research. The variational iteration method is employed to develop a closed form semi-analytical solution. In comparison with the available numerical solution procedures, convergence of the present solution scheme is much higher and the solution algorithm converges unconditionally. Therefore, novelties may be found in the treated subject, modeling procedure, and solution algorithm. Comparisons made with results of the available numerical procedures, reveal the efficiency and accuracy of the presented solution algorithm.

Thickness effect on the contact behavior of a composite laminate indented by a rigid sphere

A modified Hertz contact law is proposed for the contact problem of a laminated plate indented by a rigid sphere, which involves the effects of the thickness, in-plane dimensions and boundary conditions of the plate. The model shows that the deflection difference between the center and the edge of the contact area is a key factor dominating the effects of these variables on the force–indentation response. The modified contact law is assumed to follow the same mathematical formulation as the Hertz law. However, the contact stiffness is no longer constant but varies as a function of the contact force due to the thickness effect. The predictions of the force–indentation response show improved agreement over early work by Yang and Sun. Over the present analysis range, the thickness effect may be negligible if the plate thickness is greater than 2 mm. On the other hand, the thickness has a more significant effect on the force–indentation response if the plate thickness is less than 2 mm. Moreover, the in-plane dimensions and the boundary conditions of the plate show little influence on the force–indentation response within the present scope of analysis.

Response of composite beam under low-velocity impact of multiple masses

A theoretical method is presented to analyse the low-velocity impact dynamics of a system which consists of a laminated beam and multiple impactor masses. The contact forces between the beam and the impactors are treated as the internal forces of the system. The beam displacement field is defined using a higher-order shear deformation theory and the interaction behween the beam and the impactor masses is described with the modified Hertzian contact law. The potential and kinetic energies of the system (the energies associated with the beam and the impactors) are formulated in terms of the displacement field of the beam and the modified contact law. These are substituted into the Lagrangian equation to derive the governing equations of the system, which are nonlinear second-order differential equations. Simple polynomials are assigned to the shape functions of the beam in order to satisfy arbitrary boundary conditions. The present method is general in that it allows the impactors to be of different masses striking the beam with different initial velocities at arbitrary locations on the top surface of the beam. There is no limitation on the number of impactors that can be considered.

Low-velocity impact response for laminated stepped beam

A numerical method is presented to evaluate the response of a laminated stepped beam under a transverse low velocity impact. The impactor and the laminated beam are considered as a single system and thus the impact force becomes the internal force of the system. The energies associated with each span of the beam and the impactor mass are derived in the form of a higher order shear deformation theory and Sun's modified contact law. These are substituted in the Lagrangian equation of motion to arrive at the governing equations of the system. The equations are uncoupled by principal coordinate transformations and then solved numerically. The solutions can be obtained for arbitrary boundary conditions. Natural frequency parameters are compared with the available literature results wherever possible and very good agreement is achieved. New case studies are discussed in detail to demonstrate the efficiency of the method.

Geometrically nonlinear impact response of thin laminated imperfect cylindrical panels

Although a large number of studies dealing with the impact response of flat laminated panels has been conducted in the past, relatively little work seems to be available on impact of laminated cyclindrical panels. In this paper an analytical study is conducted to predict the nonlinear transient response of thin imperfect laminated cylindrical panels subjected to impact loads. A review is also given regarding the state of the art on the impact response of laminated plates and shells. The imperfect panels are modeled using a 48 degree-of-freedom (DOF) finite element based on the classical laminated plate theory. Linear and nonlinear transient responses are obtained using the modal superposition method (MSM) and a reduction method based on Ritz vectors (R-R), both in conjunction with direct integration schemes (Wilson- θ or Newmark methods). The effect of the number of modes used is also discussed. A modified contact law is incorporated to evaluate the impact loads due to a projectile. Different nondimensional shell radii [ r/h] are used in order to study the effect of nondimensional shell radii [ r/h] on the impact response of laminated cylindrical panels. The effect of geometric imperfections on linear and nonlinear transient response under sudden impact loads is also analyzed. Both the MSM and R-R methods used in the analysis are found to have acceptable accuracy and the accuracy increased with the number of modes used. It is observed that the reduction of the nondimensional shell radius [ r/h] causes significant changes in the impact response of the cylindrical panel: a significant decrease in the central deflection and contact force histories. The introduction of geometric imperfections led to similar conclusions.

Response of Composite Laminates to Contact Loads and Relationship to Low-Velocity Impact

Experimental results are presented for contact and low-velocity-impact re sponses of composite laminates by rigid spheres. The contact phenomenon is different in small and large indentation stages, which is due to occurrence of laminate damage. The contact force corresponding to the onset of laminate damage is found to be independent of the indentor size. In the small indentation stage where the plate is intact, the change of laminate stacking sequence has an insignificant effect on the force-indentation relation ship. However, the force-indentor size relationship does not follow the modified contact law for thin laminates indented by large indentors. Beyond the small indentation stage, damage occurs. The size of the delamination area is proportional to the applied loads and the number of repeated loading cycles. The passage of the delamination crack front can be detected using strain gauges. Also, the indentation spring becomes stiffer under repeated loading. Finally, the delamination damage and the strain history in a laminate at low- velocity impact are found to be very similar to those in the contact test except that vibra tion was observed in the impact test.