Project work in Continuum Mechanics, FMEN2 Evaluation of material model for calculating pressure and speed in a granular flow Introduction In continuum mechanics we always consider materials as continuum, even though the real material is discontinuous. Granular materials as sand, gravels and different powders are examples of material consisting of discrete particles, but successfully modeled by continuum mechanics approaches. In this project work we will study a granular flow along an inclined plane, see Fig.. The gravity is assumed to be the only driving force of the flow. The granular particle are assumed to be rigid i.e. incompressible and the voids between the particle are filled with air. The volume of the voids can vary during the flow process. To describe this we usually introduce the solid fraction v, defined as the fraction of solid material per unit volume. The solid fraction v and the velocity U downward the leaning plane are determined from experiment and plotted in Fig.2. g Rigid wall y θ v Free surface x h=.3 m Figure : Schematic description of a granular flow downward an inclined plane. LTH, Box 8, S-22 LUND Sweden Tel: +46 ()46 222339 Fax: +46 ()46 222462 Aylin.Ahadi@mek.lth.se www.mek.lth.se
free surface free surface Figure 2: Experimental result for a granular flow downward the leaning plane. (Adopted from Savage, S.B., Advances in Applied Mechanics, vol. 24, p. 289-366, 984) Constitutive relations In order to determine the flow of the granular material one can make use of the balance equation of momentum, Assume an acceleration free flow i.e. x = x = divt + b () In order to have a closed system of equations i.e. the same number of unknowns and number of equations, a constitutive relation for the stress tensor, T must be introduced. In this project we will evaluate two different constitutive relations for the stress tensor. Both relations are of first order differential type. LTH, Box 8, S-22 LUND Sweden Tel: +46 ()46 222339 Fax: +46 ()46 222462 Aylin.Ahadi@mek.lth.se www.mek.lth.se 2
Constitutive relation The constitutive relation to be evaluated is the constitutive assumption for a linear viscous fluid. The granular flow is assumed to behave as a linear viscous fluid and there is no dependence on the density, i.e. the flow is incompressible. The stress tensor in this case is given by, T = - p + λ tr D+ 2µ D (2) where p is the pressure and µ, λ are material constants. The rate of deformation tensor D is defined by, D= x x y x z x + y x + 2 2 z x y x y y z + + x y y z y 2 2 z x z y z + + x z y z 2 2 z (3) The gradient of the density is denoted by v, Assumptions v = grad (4) The inclination of the plane is 3 degreesθ = 3, and the plane and the fluid are assumed to be infinitely extended in x- and y-directions. This leads to an assumption for the velocity field and the density distribution depending on the y coordinate only, x = xy ( ), = ( y) (5) The pressure p is also assumed to vary in y-direction only i.e. p = p(y). The only body force acting on the flow is the gravity i.e. b = (gx, gy, ). LTH, Box 8, S-22 LUND Sweden Tel: +46 ()46 222339 Fax: +46 ()46 222462 Aylin.Ahadi@mek.lth.se www.mek.lth.se 3
The equations of motions () and the constitutive relation given by (2) lead to two equations with 3 unknown quantities - the density = ( y), the velocity x = xy ( ) and the pressure p = p(y). Using the experimental results presented in Fig.2, an expression for one of the unknown quantities can be determined. The experimental results for the second quantity can be used as an evaluation. In this project we assume that the distribution of the density is known from the experiment and by determining an analytical expression for = ( y) one can determine the gradient of the density v, v = grad. Then by use of the equations of motion the velocity field x = xy ( ) and the pressure p = p(y) can be calculated and compared to experimental data to validate the results. The density is assumed to be distributed as follows, y y L - L h h y = y h = e e ( / ) (A +B +C +D) h (6) where A,B,C,D and L are constants and is the density of the granular grains. Assume =. Introducing the volume fraction ν the last equation can be rewritten as, y y L - L y h h ν ( y) = = (A e +B e +C +D) (7) h The following boundary conditions must been fulfilled, ν( y/ h= ) = ν y ν( y/ h=.3) = ν ν( y/ h= ) = νy ν ( y/ h =.3) = 3 y, < y/ h< (8) whereν y is the volume fraction at y=, ν 3 y is the maximum volume fraction at y/h=.3 ν is the volume fraction at y/h=. ν ( y/ h=.3) is the derivative with respect to y. and y ν y, ν y and ν 3 y can be read from Fig.2b. LTH, Box 8, S-22 LUND Sweden Tel: +46 ()46 222339 Fax: +46 ()46 222462 Aylin.Ahadi@mek.lth.se www.mek.lth.se 4
Exercise The constant A, B, C and D in equation (7) can be evaluated from the boundary conditions. Determine these constants as a function of L and plot the density function for different values of L varying between and 2. Choose the value of L that fits the curve best to the experimental data on Fig.2b. Then calculate the final values for the constants A, B, C and D. Exercise 2 With the final expression for the density determined in Exercise, now we have a close system of equation and the velocity field and the pressure can be determined. Separate the equation of motion () into equations in x- and y-direction. Use the first constitutive relation i.e. the linear viscous assumption and insert that into the equation of motion. Calculate the pressure p from the equation of motion in y-direction and the velocity field x = xy ( ) from the equation of motion in x-direction. Introduce and read from the Fig.2a suitable boundary conditions to determine the final expressions. The material parameter µ can be used to scale the velocity field or to change its direction with µ =-. Use the value µ =- and plot the velocity field. Exercise 3 Consider the second constitutive relation, which is an isotropic. The stress tensor depends linearly on the symmetric part of the rate of deformation tensor, D, and quadratically on the gradient of the density v. The constitutive relation is given by, T = - p + λ tr D+ 2 µ D+ µ v v + µ ( v Dv + Dv v ) 2 where p is the pressure and λ, µ, µ and µ 2 are material constants. Separate the equation of motion into equations in x- and y-direction. Calculate the pressure p from the equation of motion in y-direction and x y from the equation of motion in x-direction. GOOD LUCK! Aylin LTH, Box 8, S-22 LUND Sweden Tel: +46 ()46 222339 Fax: +46 ()46 222462 Aylin.Ahadi@mek.lth.se www.mek.lth.se 5