Parallel RBF-FD solution of the Boussinesq's problem

2019 ◽  
Author(s):  
G. Kosec ◽  
J. Slak
Keyword(s):  
Boussinesq’S Problem

Boussinesq's problem for a flat-ended cylinder

1946 ◽  
Vol 42 (1) ◽  
pp. 29-39 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Free Surface ◽  
Rigid Body ◽  
Elastic Medium ◽  
Normal Displacement ◽  
Full Account ◽  
Elastic Stresses ◽  
Special Cases ◽  
Cylindrical Punch ◽  
Distribution Of Stress ◽  
Boussinesq’S Problem

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

On Boussinesq's problem

2001 ◽  
Vol 39 (3) ◽  
pp. 317-322 ◽  
Author(s):  
A.P.S. Selvadurai
Keyword(s):  
Boussinesq’S Problem

scholarly journals Solution of the Boussinesq’s problem in evaluating ratio between shear stress and contact pressure

2020 ◽  
pp. 139-151
Author(s):  
Е.Г. Хитров ◽  
А.В. Андронов ◽  
Е.В. Нестерова
Keyword(s):  
Shear Stress ◽  
Stress Distribution ◽  
Normal Stress ◽  
Contact Pressure ◽  
Stress Function ◽  
Average Pressure ◽  
Normal Stress Distribution ◽  
Contact Patch ◽  
Wide Range ◽  
Boussinesq’S Problem

Решение фундаментальной задачи Буссинеска широко используется в технических науках и позволяет эффективно решать широкий спектр задач науки о лесозаготовительном производстве. На его основе удается получить практически значимые результаты в области оценки распределения напряжений, возникающих в обрабатываемом материале под воздействием рабочего органа. Цель нашего исследования - проанализировать результаты расчетов и установить соотношение максимального значения касательного напряжения и среднего значения давления по пятну контакта рабочего органа с обрабатываемом материалом. Теоретическую основу работы составляют уравнения распределения нормальных и касательных напряжений, возникающих в упругом полупространстве при вдавливании в него жесткого клина. В результате анализа теоретических расчетов показано, что характер затухания нормального напряжения по глубине деформируемого массива материала с высокой точностью аппроксимируется квадратичной функцией (на основе полученной приближенной функции выполнено сопоставление среднего давления по пятну контакта индентора с массивом и нормального напряжения по глубине массива). При этом, как показали результаты расчетов, функция распространения касательного напряжения в деформируемом массиве имеет экстремум. Выполнено сопоставление полученных данных по значению экстремума функции касательного напряжения со значением приближенной функции нормального напряжения на границе контакта индентора сдеформируемым массивом. В результате показано, что максимальное по модулю касательное напряжение составляет 11-12% среднего контактного давления. Расчеты проведены при варьировании коэффициента Пуассона материала массива, установленное соотношение остается практически неизменным. Solution of fundamental Boussinesq’s problem is widely used in technical sciences and allows effectively solving a wide range of problems in forestry science. On its basis, it is possible to obtain practically significant results in the field of assessing the distribution of stresses arising in processed material under the influence of a working body. The purpose of our study is to analyze the results of calculations and establish the ratio of the maximum value of the shear stress and the average pressure over the contact patch of the working body with the material being processed. The theoretical basis of the work is formed by the equations for the distribution of normal and tangential stresses arising in an elastic half-space when a rigid cone is pressed into it. As a result of the analysis of the results of theoretical calculations, it was shown that the character of the normal stress distribution over the depth of the deformed massif of material is approximated with high accuracy by a quadratic function (based on the obtained approximate function, the average pressure over the contact patch of the indenter with the massif and the normal stress over the depth of the massif were compared). In this case, as shown by the results of calculations, the function of the shear stress distribution in the deformed massif has the extremum. Comparison of the obtained data on the value of the extremum of the shear stress function with the value of the approximate normal stress function at the interface of the indenter contact with the deformable mass is performed. As a result, it is shown that the maximum shear stress in absolute value is 11-12% of the average contact pressure. The calculations were carried out with varying Poisson's ratio of the massif material; the established ratio remains practically unchanged.

Effects of a Change of Poisson’s Ratio Analyzed by Twinned Gradients: Illustrated by Simple Solutions of Boussinesq’s Problem of a Normal Force and Cerruti’s Problem of a Tangential Force Acting on the Surface of a Large Solid, and by a Simple Derivation of the Stresses in a Rotating Thick Disk

1940 ◽  
Vol 7 (3) ◽  
pp. A113-A116
Author(s):  
H. M. Westergaard
Keyword(s):  
Poisson’S Ratio ◽  
Normal Force ◽  
Plane Surface ◽  
Tangential Force ◽  
Thick Disk ◽  
Poisson's Ratio ◽  
Total Force ◽  
Principal Stresses ◽  
Simple Derivation ◽  
Boussinesq’S Problem

Abstract Some problems of elasticity have a simple solution for a particular value of Poisson’s ratio. For example, Boussinesq’s problem of a normal force and Cerruti’s problem of a tangential force, acting on the plane surface of a semi-infinite solid, are solved when Poisson’s ratio is 1/2 by referring to Kelvin’s problem of a force at a point in the interior of an infinite solid. For, when Poisson’s ratio is 1/2, the solution of Kelvin’s problem can be stated in terms of one principal stress at each point, acting along the radial line from the point of the load; the other principal stresses are zero; and one half of the total force may be assigned to one half of the infinite solid. For other values of Poisson’s ratio terms must be added in the formulas for the displacements and stresses. The derivations that have been available are somewhat lengthy, especially for Cerruti’s problem. The difficulties are reduced by a simple analytical device, here called “the twinned gradient.” The displacement to be added by the change of Poisson’s ratio is stated as the gradient of a potential except that one of the components is replaced by its twin, an identical component in reversed direction. This device also lends itself to a simplification of the analysis of stresses in a rotating thick disk.

Sign in / Sign up

Export Citation Format

Share Document