scholarly journals Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials

2015 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
Mohamed A. Agwa ◽  
António Pinto Da Costa
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Initial Conditions ◽  
Orthotropic Materials ◽  
Isotropic Case ◽  
Linear Elastic ◽  
Analytic Expressions ◽  
The Coefficient Of Friction ◽  
Unstable Solutions ◽  
Computer Algebra Software

Abstract The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson’s ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.

scholarly journals ELASTIC MOMENT HALF-SPACE UNDER THE ACTION OF AXISYMMETRIC NON-STATIONARY SURFACE KINEMATIC PERTURBATIONS

2019 ◽  
Vol 81 (1) ◽  
pp. 40-52
Author(s):  
Le Thai Tran ◽  
D. V. Tarlakovskii
Keyword(s):  
Half Space ◽  
Initial Conditions ◽  
Influence Functions ◽  
Moment Stress ◽  
Plane Normal ◽  
Cylindrical Coordinate ◽  
Analytic Expressions ◽  
The Moment ◽  
Relationship Of ◽  
Angle Of Rotation

The article deals with elastic homogeneous isotropic half-space filled with the Cosserat medium. A cylindrical coordinate system is used. A closed system of equations includes equations for the non-trivial components of the displacement and rotation potentials, as well as relations relating the displacements to the potentials and components of the stress and moment stress tensors with displacements and the angle of rotation. On the boundary plane, normal displacements are specified, and the angle of rotation and tangential displacements are absent. Initial conditions are zero. All components of the stress-strain state are assumed to be limited. A system of dimensionless quantities is used. The solution is represented as a convolution with respect to time and a generalized convolution with respect to the radius of surface perturbations with influence functions. To construct them, Hankel transforms are applied along the radius and Laplace in time, as well as expansion in power series in a small parameter in the linear approximation. Found image of all surface influence functions. For example, the following function is considered corresponding to the normal voltage. Its original is on the border of a half-space using the connection of axisymmetric and plane problems, namely, taking into account the proportionality of the Hankel and Fourier images respectively. It uses the previously constructed solution of the planar problem. As a result, the desired function is represented as integrals, which are understood in the sense of regularized values. Analytic expressions for these integrals and the influence function itself are found. Examples of calculations of the influence functions of a granular composite of aluminum shot in an epoxy matrix are given. Two variants of action on the half-space of surface normal displacements are also considered: the disturbance concentrated at the origin and distributed in a circle. Analysis of the results shows that the effect of the moment properties of the medium depends significantly on the value of the parameter characterizing the relationship of elastic displacements and rotational motions. For the material in question in quantitative terms it is not great. However, consideration of moment stress leads to qualitative changes. Namely, there is an additional wave front.

scholarly journals Satisfying Boundary Conditions in Homogeneous, Linear-Elastic and Isotropic Half-Spaces Subjected to Loads Perpendicular to the Surface: Distributed Loads on Adjacent Contact Areas

2019 ◽  
Vol 6 (1) ◽  
pp. 11-29 ◽  
Author(s):  
Elena Ferretti
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Closed Form Solution ◽  
External Pressure ◽  
Second Order ◽  
Experimental Program ◽  
Contact Areas ◽  
Linear Elastic ◽  
Order Solution ◽  
Homogeneous Linear

AbstractThis work originates from an experimental program on strain distribution near the loaded surface of an airfield concrete pavement,which provided us with results that contrast with the rheological predictions of Boussinesq for a homogeneous, linear-elastic and isotropic half-space. We already reviewed and extended the original work carried out by Boussinesq in previous papers, to provide a closed form second order solution that enabled us to establish a good match between analytical and experimental findings for point-loads. In this paper, we have explained why Boussinesq’s closed form solution for a homogeneous linear-elastic and isotropic half-space subjected to a point-load is not exact, as believed until now, but approximated. Then, we have shown that our second order solution is the actual solution of Boussinesq’s problem. We have also presented the numerical analysis of second order for rectangular and elliptical contact areas, both loaded by uniform and parabolic laws of external pressure. Moreover, we have evaluated the interaction effect provided on the surface of a concrete half-space by the twin wheels of an aircraft landing gear. Extension of the solution to layered systems is also possible, for improving the knowledge of stress propagation into airfield pavements and promoting more effective design standards.

Dynamic Interaction of Two Independent SDOF Oscillators Supported by a Flexible Foundation Embedded in a Half-Space: Closed-Form Analytical Solution for Incident Plane SH-Waves

2021 ◽  
pp. 1-24
Author(s):  
Liguo Jin ◽  
Liting Du ◽  
Haiyan Wang
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Half Space ◽  
Strong Interaction ◽  
Rigid Foundation ◽  
Structural Stiffness ◽  
Sh Waves ◽  
Closed Form Analytical Solution ◽  
Flexible Foundation ◽  
Plane Sh Waves

This paper presents a closed-form analytical solution for the dynamic response of two independent SDOF oscillators standing on one flexible foundation embedded in an elastic half-space and excited by plane SH waves. The solution is obtained by the wave function expansion method and is verified by comparison with the results of the special cases of a rigid foundation and the published research result of a flexible foundation. The model is utilized to investigate how the foundation stiffness influences the system response. The results show that there will be a significant interaction between the two independent structures on one flexible foundation and the intensity of the interaction is mainly dependent on foundation stiffness and structural stiffness. For a system with more flexible foundation, strong interaction will exist between the two structures; larger structural stiffness will also lead to a strong interaction between the two structures. When the structural mass and the structural stiffness are all larger, the flexible foundation cannot be treated as a rigid foundation even if the foundation stiffness is many times larger than that of soil. This model may be useful to get insight into the effects of foundation flexibility on the interaction of two independent structures standing on one flexible foundation.

Exact Analysis of Dynamic Sliding Indentation at any Constant Speed on an Orthotropic or Transversely Isotropic Half-Space

2002 ◽  
Vol 69 (3) ◽  
pp. 340-345 ◽  
Author(s):  
L. M. Brock
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Plane Strain ◽  
Half Space ◽  
Contact Zone ◽  
Transversely Isotropic ◽  
Exact Formula ◽  
Constant Speed ◽  
Displacement Fields ◽  
Rigid Indentor

A plane-strain study of steady sliding by a smooth rigid indentor at any constant speed on a class of orthotropic or transversely isotropic half-spaces is performed. Exact solutions for the full displacement fields are constructed, and applied to the case of the generic parabolic indentor. The closed-form results obtained confirm previous observations that physically acceptable solutions arise for sliding speeds below the Rayleigh speed, for a single critical transonic speed, and for all supersonic speeds. Continuity of contact zone traction is lost for the latter two cases. Calculations for five representative materials indicate that contact zone width achieves minimum values at high, but not critical, subsonic sliding speeds. A key feature of the analysis is the factorization that gives, despite anisotropy, solution expressions that are rather simple in form. In particular, a compact function of the Rayleigh-type emerges that leads to a simple exact formula for the Rayleigh speed itself.

A TWO-DIMENSIONAL STATIC DEFORMATION OF AN IRREGULAR INITIALLY STRESSED MEDIUM

2008 ◽  
Vol 22 (20) ◽  
pp. 3473-3485
Author(s):  
M. M. SELIM
Keyword(s):  
Closed Form ◽  
Initial Stress ◽  
Half Space ◽  
Static Deformation ◽  
Initial Stresses ◽  
Two Dimensional ◽  
Eigenvalue Approach ◽  
Line Load ◽  
Notable Effect ◽  
Approach Method

The paper discusses the problem of a two-dimensional static deformation as the result of normal line-load acting inside an irregular initially stressed isotropic half-space. The eigenvalue approach method has been used. The irregularity is expressed by a rectangle shape. Further, the results for the displacements and stresses have been derived in the closed form. The effect of initial stress and irregularity are shown graphically. It was found that the initial stresses as well as irregularity have a notable effect on this deformation.

Effect of Surface Roughness on Adhesion and Friction of Microfibers in Side Contact

2008 ◽  
Author(s):  
Mircea Teodorescu ◽  
Carmel Majidi ◽  
Homer Rahnejat ◽  
Ronald S. Fearing
Keyword(s):  
Mathematical Model ◽  
Surface Roughness ◽  
Closed Form ◽  
Contact Mechanics ◽  
Elastic Contact ◽  
Linear Elastic ◽  
Multi Scale ◽  
Analytic Approximations ◽  
Effect Of Surface

A multi-scale mathematical model is used to study the effect of surface roughness on the adhesion and friction of microfibers engaged in side contact. Results are compared to closed-form analytic approximations derived from linear elastic contact mechanics.

Closed-Form Relations for Stress Intensity Factor Influence Coefficients for Axial Outside Surface Flaws in Cylinders for Appendix A of ASME Section XI

2016 ◽  
Author(s):  
Steven X. Xu ◽  
Darrell R. Lee ◽  
Douglas A. Scarth ◽  
Russell C. Cipolla
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Closed Form ◽  
Inside Surface ◽  
Linear Elastic ◽  
Evaluation Procedures ◽  
Influence Coefficients ◽  
Elastic Fracture ◽  
Flaw Location

Linear elastic fracture mechanics based flaw evaluation procedures in Section XI of the ASME Boiler and Pressure Vessel Code require calculation of the stress intensity factor. Article A-3000 of Appendix A in ASME Section XI prescribes a method to calculate the stress intensity factor for a surface or subsurface flaw by making use of the flaw location stress distribution obtained in the absence of the flaw. The 2015 Edition of ASME Section XI implemented a number of significant improvements in Article A-3000, including closed-form equations for calculating stress intensity factor influence coefficients for circumferential flaws on the inside surface of cylinders. Closed-form equations for stress intensity factor influence coefficients for axial flaws on the inside surface of cylinders have also been developed. Ongoing improvement efforts for Article A-3000 include development of closed-form relations for the stress intensity factor coefficients for flaws on the outside surface of cylinders. The development of closed-form relations for stress intensity factor coefficients for axial flaws on the outside surface of cylinders is described in this paper.

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