Solid bonding

Jean-Yves Delenne

1 Introduction

The cohesion of granular materials reflects their ability to resist external tensile stresses. This property enhances also the shear strength that is used to assess the cohesion of granular soils from shear tests performed at different confining pressures. The microscopic origin of cohesion is the presence of attraction forces between particles. These interactions, to which we refer as cohesive interactions, have different physical and chemical origins. They manifest themselves between two particles by resistance to separation, shear or rolling. The phenomenology of simple interactions can be recovered by integrating the electromagnetic forces acting on very smaller scales. This upscaling procedure is, however, rather complex. Moreover, most properties of granular media emerge from their many-body feature and the dissipative nature of their interactions. For this reason, the relevant scale for modeling granular materials is often that of the particles and the cohesive interactions need to be considered at this scale.

DEM algorithms can easily be extended to account for cohesive interactions which are often simply supplemented to the repulsive elastic and frictional interactions of cohesionless materials. We present here a general framework for the implementation of cohesion in a discrete-element framework. Generally, simple models of interaction laws are privileged in DEM so as to reduce time-consuming operations arising from too many elementary operations. In any case, one has to check the impact of the simplifications on the resulting global behavior of the material.

2 Model of contact

In the case of a non-frictional contact, the force acting between two particles is normal to the contact plane and can be additively decomposed into three forces:

fn = fen + fdn + fcn
(1)

where fne is the repulsive force at contact, fnd a viscous force, and fnc the cohesion force.

(a) PIC   (b) PIC

Figure 1: Hertz contact. (a) Circular contact zone of diameter 2a and contact deflexion δn. (b) Evolution of the contact force as a function of contact deflexion.

For spherical particles, the contact surface takes the form of a disk of radius a at which the pressure is not uniformly distributed; figure 1. We make the following assumptions about the contact:

Under these assumptions, the contact force is given by the Hertz contact law (Maugis [2000]):

 e   4 ⋆√ ---    3∕2
fn = 3E   R⋆(- δn)
(2)

avec 1-
E⋆ = 1--ν21-
E1 + 1-ν22
 E2 where E1 et E2 are the Young modulus of the two particles and ν1 and ν2 the poisson ratio and R is the harmonic mean of the particles radii.

For cohesive granular materials, the effect of cohesion often dominates that of confining stresses. The macroscopic behavior is much more dictated by the cohesive interactions than by the non-linear behavior at the contact. Therefore, a linear approximation of the contact force is relevant:

 e
fn = kn (- δn)
(3)

where kn is the contact normal stiffness.

3 Models of cohesive bond

The mechanical approach presented here in a 2D configuration (Delenne et al. [2002]) is a Lagrangian approach where the kinematics and deformations are described from a reference state. The presence of cohesion limits the motion so that the displacements are more relevant than velocity as kinematic variables.

The kinematics is described in a global reference frame (O,⃗x,⃗y). We introduce two material points I1 and I2 belonging respectively to particle 1 (center C1 and radius R1) and particle 2 (center C2 and radius R2); figure 2a and 2b. The reference state is characterized by the coordinates of the centers of particles and their radii. The point of cohesion between two particles 1 and 2 is located by the angle α12.

(a)PIC (b)PIC

Figure 2: Kinematic parameters: (a) initial state (θ1 = 0); (b) deformed state.

At the reference state, the points I1 and I2 coincide; figure 2a. Subsequently, these two points are tracked in their movements through the movements and the rotations (θ1 and θ2) of the two particles. The normal δn and tangential δt displacements and the rotation θ around the cohesion point is calculated in the local frame (⃗n,⃗t) (where ⃗n = ---→
C2C1  ∥∥---→∥∥
∥C2C1∥ and ⃗t is a unit vector orthogonal to ⃗n):

    ∥∥---→ ∥∥
δn = ∥C1C2 ∥- (R1 + R2 )
(4)

    --→  -→
δt = I1I2 ⋅ t
(5)

θ = θ1 - θ2
(6)

(a) PIC (b) PIC (c) PIC

Figure 3: Degrees of freedom at the local scale (relative displacement at contact): (a) normal displacement δn; (b) tangential displacement δt; (c) rotation θ.

The cohesion is taken into account through local relationships between the displacements (δn,δt,θ) and the contact force (fn,ft,M) where M is the torque at contact. In general, these relationships can be expressed as a cohesion law ψ:

(fn,ft,M ) = ψ(δn,δt,θ)
(7)

A local failure criterion governs the transition between the cohesion state and the sliding-contact state. This criterion is expressed through a yield function:

κ (fn,ft,M )rupt = 0
(8)

Different failure criteria may be used depending on the nature of the interface. Figure 4a shows the case of a failure criterion only based on in tensile strength. In this case, we consider that no rupture occurs in pure shear or torsion. This extreme case is similar to the case of normal adhesion for which the action of cohesion is only normal to the contact.

In the case of cemented particles, the failure may occur in tension, but also under shearing and bending. Moreover, for particles that are not in contact because of the presence of an interposed thick binder, fracture may occur in compression. In this case it may be necessary to use different failure thresholds for the different degrees of freedom; figure 4b. It may also be necessary to take into account the cross effects due to combined loading such as compression plus shearing or tension plus torque. This can be done by using more complex criteria of elliptical shape, for example; figure 4c.

(a) PIC

(b) PIC

(c) PIC

Figure 4: Example of failure criteria for a cohesive bond.

PIC

Figure 5: Failure criterion of paraboloid shape.

To test the relevance of this approach, the cohesion law and failure criteria were determined experimentally for a 2D model material consisting of aluminum cylinders glued together with an epoxy resin (Delenne et al. [2004]). Tensile, compression and shear forces as well as torques were applied on a pair of two cylinders with a cohesive bond. For this model medium, these experiments show that the compressive failure threshold is very high compared to the threshold in tension so that the threshold in compression may be set to infinity (Figure 5). A paraboloidal yield surface was found to fit the data and used in the numerical simulations:

    (-ft)2  ( M--)2   fn
κ =  ftc  +   M c   + fcn - 1
(9)

For κ < 0, there is cohesion bond, otherwise the bond fails and the contact becomes frictional without cohesion. In the case of an irreversible failure (no rejoining), one can use a parameter η which describes the damaged state of the bond (Delenne et al. [2008]). Another possibility is to modify the yield surface, corresponding to transition from elastic behavior to plastic behavior with a hardening parameter.

PIC

Figure 6: Adhesion model with hysteresis.

Luding et al. proposed a model for plastic contact as shown in figure 6. This model takes into account the normal direction but can easily be extended to more complex loading paths:

     (
     {        - k1δn si - k2(δn - δ0) ≥ - k1δn
f n =   - k2(δn - δ0) si - k1δn > - k2(δn - δ0) > kcδn
     (          kcδn si kcδn ≥ - k2(δn - δ0)
(10)

where k1 < k2. Note that for an initial load, the force increases linearly with the deflexion δ to a maximum value δmax. The maximum value is stored as a memory variable. The line of slope k1 is the limit value of force for a given value of δ. During unloading, the force declines along the line of slope k2 down to 0 for values of δ which vary from δmax to δ0 = (1- k1∕k2)δmax. The value δ0 represents a plastic deformation at contact. Upon reloading, the force increases following the line of slope k2 to the maximum value already reached δmax and the follows the line of slope k1. Unloading for values below f0 (defined for δ = δ0) leads to an attraction force. This force may decrease until fmin for a value δmin = (k2 - k1)δmax(k2 + kc) along the slope k2. For δ < δmin, the force follows the slope -kc.

References

   J. Y. Delenne, M. S. El Youssoufi, and J. C. Bénet. Comportement mécanique et rupture de milieux granulaires cohésifs. CRAS mécanique, 330: 475–482, 2002.

   Jean-Yves Delenne, Moulay Saïd El Youssoufi, Fabien Cherblanc, and Jean-Claude Benet. Mechanical behaviour and failure of cohesive granular materials. International Journal for Numerical and Analytical Methods in Geomechanics, 28(15):1577–1594, 2004. URL http://dx.doi.org/10.1002/nag.401.

   J.Y. Delenne, Y. Haddad, J.C. Bénet, and J. Abecassis. Use of mechanics of cohesive granular media for analysis of hardness and vitreousness of wheat endosperm. Journal of Cereal Science, 47(3):438–444, 2008.

   D. Maugis. Contact, Adhesion and Rupture of Elastic Solids. Springer, 2000.