Investigation of influence of baffles on landslide debris mobility by 3D material point method

Abstract

In mountainous terrain, landslide debris is a common occurrence around the world that can potentially result in catastrophic consequences to downstream residents and facilities. Arrays of baffles are increasingly used as energy dissipaters for the protection from debris flow due to their low cost and environmental impact. However, the development of a numerical tool for the rational design of such structures is still a challenge. In this work, a material point method computational framework is presented, using two contact models to describe the landslide debris movement and interaction with structures, respectively. Flume model experiments are adopted as calibration and good agreements are exhibited in flow kinematics between computed results and physical model tests. Simulations on an idealized scenario of landslide debris flow show how the baffle geometry/arrangement has key effects on the debris movement and deposition. The impact force in soil-structure interaction has also been studied, showing the ability of this method on evaluation force characteristics acted on baffle structures.

Appendix. MPM computational cycle

A computational cycle of the MPM solution generally consists of four steps.

• 1. In each time step, particle mass and momenta along with other quantities needed for algorithms are extrapolated to nodes on the mesh grid. The mass of an arbitrary node i is given by

$$ {m}_i=\sum \limits_{p=1}^n{S}_{ip}{m}_p $$

(23)

where S_ip is the shape function of node i for particle p, m_p is the mass of a material point.

The nodal momentum is given by

$$ {\mathrm{p}}_i=\sum \limits_{p=1}^n{S}_{ip}{m}_p{\mathrm{v}}_p. $$

(24)

where v_p is velocity of the material point.

• 2. The particle strains are calculated based on the nodal velocity field, and are used to update the particle stresses by the constitutive model. Then the nodal internal force and external force (neglecting surface tractions) at the i^th grid node are calculated by

$$ {\mathrm{f}}_i^{\mathrm{int}}=-\sum \limits_p\varDelta {S}_{ip}\cdot {\upsigma}_p{V}_p $$

(25)

$$ {\mathrm{f}}_i^{\mathrm{ext}}=\sum \limits_p{S}_{ip}{\mathrm{b}}_p{m}_p $$

(26)

where b_p is the body force acting on the p^th particle, V_p is the particle volume, σ_p is the Cauchy stress at the particle, and ΔS_ip is the gradient of the i^th grid shape function over the p^th particle domain. From the forces, the updated nodal velocities are computed,

$$ {\overline{\boldsymbol{v}}}_i^{n+1}={\boldsymbol{v}}_i^n+\frac{{\boldsymbol{f}}_i^{\mathrm{int}}+{\boldsymbol{f}}_i^{\mathrm{ext}}}{m_i}\Delta t $$

(27)

In Eq. (5), the updated velocity, ${\overline{\boldsymbol{v}}}_i^{n+1}$, neglects the effect of contact forces. As described in the following section, contact forces modify the velocity field to compute a corrected ${\mathrm{v}}_i^{n+1}$ from ${\overline{\boldsymbol{v}}}_i^{n+1}$.

• 3. The updated grid velocity field is used to compute the updated velocity at the particle

$$ {\mathrm{v}}_p^{n+1}={\mathrm{v}}_p^n+{\sum}_i{S}_{ip}\frac{{\mathrm{f}}_i^{\mathrm{int}}+{\mathrm{f}}_i^{\mathrm{ext}}}{m_i}\varDelta t $$

(28)

• 4. Update material point positions by

$$ {\mathrm{x}}_p^{n+1}={\mathrm{x}}_p^n+{\sum}_i{S}_{ip}{\mathrm{v}}_i^{n+1}\varDelta t $$

(29)

After the update, background mesh can be discarded or reused for the next time step in its initial, undistorted form. The previously mentioned procedure is repeated for each incremental step.