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Full understanding the interaction mechanisms between flow-like landslides and the impacted protection structures is an open issue. While several approaches, from experimental to numerical, have been used so far, it is clear that the adequate assessment of the hydromechanical behaviour of the landslide body requires both a multiphase and large deformation approach. This paper refers to a specific type of protection structure, namely a rigid barrier, fixed to the base ground. Firstly, a framework for the Landslide-Structure-Interaction (LSI) is outlined with special reference to the potential barrier overtopping (nil, moderate, large) depending on the features of both the flow and the barrier. Then, a novel empirical method is casted to estimate the impact force on the barrier and the time evolution of the flow kinetic energy. The new method is calibrated by using an advanced hydro-mechanical numerical model based on the Material Point Method. The validation of the empirical formulation is pursued referring to a large dataset of field evi- dence for the peak impact pressure. Both numerical and empirical methods can appropriately simulate the physical phenomena. The performance of the newly proposed empirical method is compared to the literature methods and its advantages are outlined. Keywords: Impact mechanisms, Mitigation, Structure, Material Point Method the hydro-mechanical features of the impacting flow, (ii) Introduction the geometry of the structure, and (iii) initial and bound- The interaction of flow-like landslides with rigid walls, ary conditions for the specific LSI problem. Recently the obstacles, protection structures and, more recently, sin- solid–fluid hydro-mechanical coupling and the role of the gle building or cluster of buildings have been investigated interstitial fluid in the LSI mechanisms have been con - by a variety of numerical tools. Either Discrete Element sidered. For instance, the impact behaviour of saturated Method (DEM) (Teufelsbauer et al. 2011; Leonardi flows against rigid barriers (as observed in centrifuge et al. 2016; Calvetti et al. 2017) or continuum mechan- tests) was satisfactorily simulated through MPM (Cuomo ics has been adopted. For the latter, Eulerian methods et al. 2021). Most of these approaches are very recent, were extensively applied (Moriguchi et al. 2009; Kattel and still need comprehensive validation combined with et al. 2018), but Lagrangian methods such as Smoothed- more efforts to reduce the computational cost, which is Particle Hydrodynamics (SPH) (Pastor et al. 2009) and very high once realistic simulations are pursued. Material Point Method (MPM) (Bui and Fukagawa 2013; A more traditional approach is based on (i) direct Cuomo et al. 2013; Ceccato et al., 2018) have a great observation of the impact of flow-like landslides against potential. The massive use of numerical methods is barriers, and (ii) correlation of the achieved measure- related to the inner complexity of Landslide-Structure- ments. The measurements available in the literature have Interaction (LSI) mechanisms, which are related to: (i) been mostly obtained in reduced-scale flume tests (Hübl et al. 2009; Armanini et al. 2011; Canelli et al. 2012; Ash- wood and Hungr 2016; Vagnon and Segalini 2016), or in *Correspondence: scuomo@unisa.it University of Salerno, Fisciano, Italy Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 2 of 17 some cases in full-scale flume experiments (De Natale The present work investigates the impact mecha - et al. 1999; Bugnion et al. 2012). nisms of flow-like landslides against artificial barriers In general, the reduced-scale laboratory tests have been in full-scale realistic scenarios. A Conceptual Model of extensively used to derive and to validate the empirical LSI is firstly proposed. Then, a new empirical method is formulations most used to assess the peak impact pres- casted to evaluate the peak impact horizontal force and sure in the design of protection measures against land- the reduction in the kinetic energy of the flow. The new slide (Proske et al. 2011). The existing empirical methods empirical formulation is calibrated with a set of numeri- can be classified into three groups: (i) hydro-static meth - cal results, achieved by applying the MPM approach to ods, which require only flow density and thickness for analyse the hydro-mechanical interaction of saturated evaluating the maximum impact pressure (Scotton and flows with different types of barriers. The validation of Deganutti 1997; Schieldl et al. 2013); (ii) hydro-dynamic the empirical formulation is pursued with reference to a methods, based on flow density and the square veloc - large dataset containing field evidence of impact prob - ity of the flow (Bugnion et al. 2012; Canelli et al. 2012); lems for real debris flows. Finally, the novel empirical for - (iii) mixed methods, that accounts for both the static mulation is compared with those from the literature and and the dynamic components of the flow (Arattano and its potential and limitations are discussed. Franzi 2003; Hübl et al. 2009; Armanini et al. 2011; Cui et al. 2015; He et al. 2016; Vagnon 2020). The weak point is that all the empirical formulations greatly depend on Framework empirical coefficients which are difficult to estimate in It is assumed that a flow-like landslide impacts against a the practical applications due to their wide range of vari- rigid protection barrier fixed to the base ground (Fig. 1a). ation. Common to those approaches are the following The landslide body has the following features: unitary assumptions: (i) the impact load is assumed to be totally width, length L , depth h , density of the mixture ρ , ini- 1 m transferred to the structure without any dissipation tial uniform velocity v , pore-water pressure p and fric- 0 L during the impact, and (ii) the size, stiffness and iner - tion coefficient along the base ground equal to tan ϕ . tial resistance of the artificial barrier are not considered In real cases, the barrier is often built as a reinforced (Vagnon and Segalini 2016). These assumptions generally concrete vertical wall or an embankment with a steep lead to safe assessment of the peak impact force but with inclined face at the impact side. For the sake of general- large overestimation in the barrier design. Hence, some ity, here below we consider that the barrier is a trapezoid, enhancements will be proposed in this paper on both with these geometric characteristics: bottom base B, top these topics. base b , height H , inclination of the impacted side β. Fig. 1 a Conceptual scheme for Landslide-Structure-Interaction (LSI); b large overtopping, c moderate overtopping, d no overtopping Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 3 of 17 Fig. 2 Schematic of the impact problem in the proposed empirical model The LSI problem is described through the following not move. In practice, a solution is placing a base layer timelines: initial configuration ( t ), landslide propaga- of soil with an assigned frictional resistance. tion ( t < t < t ), impact of the landslide front ( t ) , The expected LSI mechanisms, in a contest of land- 0 imp imp time of the peak impact force (T ), start of the inertial slide risk mitigation, must be distinguished among stage (T ), end of LSI ( t ). Before the landslide reaches those with large (Fig. 1b) or moderate (Fig. 1c) flow 2 f the barrier ( t < t < t ), i.e., during the propagation overtopping of the barrier and that mechanism 0 imp stage, the LSI problem is governed by the basal fric- (Fig. 1d) which makes the flow to stop behind the tional force F , which acts along the bottom of the flow barrier. Consistently with the literature (Faug 2015, ( L ) and controls the reduction in flow velocity, result- among others), it is considered that the LSI dynam- ing in a decrease of the impact forces. While the flow ics is guided by the impact velocity and the height of interacts with the barrier ( t < t < T ) , additional the protection structure relative to the flow thickness. imp 2 stresses (mostly orthogonal to the impacted surface, High flow velocity predisposes to the large (Fig. 1b) hence horizontal in many applications) are produced or moderate (Fig. 1c) overtopping depending on the at the impacted side of the barrier. Many studies (e.g., height of the barrier. On the other hand, the taller is Cui et al. 2015; Song et al. 2017) demonstrated that the the barrier the more probable is that the flow is fully total impact force–time history can be simplified as retained (Fig. 1d). However, LSI dynamics is also a triangular force impulse, from nil to the peak value guided by landslide pore pressure (Cuomo et al. 2021), and then down to a static value, usually with a rise and this issue will be considered in the following. time (T ) much shorter than the decay time (T − T ). 1 2 1 According to the Newton’s Third law of motion, the Formulation of a novel empirical method mutual impact forces (F ) between the landslide and The landslide is here schematized as a rectangular with the barrier are equal and opposite. Such mutual stress mass m , leng th L , depth h , unitary width, density ρ , 1 m makes the flow to decelerate. The evaluation of the initial velocity v and it is supposed that the landslide is impact forces applied on the inclined side of the bar- completely stopped by the barrier (i.e., v(T = 0 ). The rier ( L ) is fundamental to design the structural char- latter is assumed as fixed to the base ground and indefi - acteristics of the barrier. It is worth noting that the nitely high, thus all the landslide volume is supposed to flow may overtop the barrier during the impact, gen- be retained by the barrier (Fig. 1d). erating an additional force F on the structure, mainly Based on previous studies (Hungr et al. 1984; Scot- dependent on the flow-barrier frictional contact ton and Deganutti 1997; Kwan 2012), the peak lateral ( tan δ ). In a simplified approach, F and F could be force F (Eq. 1) exerted by the flow on the obstacle is peak 1 3 neglected. Once F , F and F are given, the constraint calculated by the sum of a dynamic component F peak ,dyn 1 2 3 reactions required at the base of the barrier can be (Fig. 2a) and a height-dependent static component computed. Thus, the ultimate strength of the founda- F (Fig. 2b), as reported in Eqs. 2–3, respectively. peak ,stat tion systems can be designed so that the barrier does Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 4 of 17 F = F + F peak peak ,dyn peak ,stat (1) E (t) = mv (t) (8) The impact period T is obtained by using the impulse F = αρ v h 2 peak,dyn m (2) theorem, since the integral over time of the impact force (i.e., the impact impulse) is equal to the variation of linear F = κρ gh (3) momentum of the landslide (Eq. 4). The left side of Eq. 4 peak,stat m can be rewritten as reported in Eq. 9, where the right side corresponds to the area subtended by the piecewise func- The empirical coefficient α has a wide range of values, tion reported in Eq. 5 and plotted in Fig. 2a. Once known ranging from 0.4 to 12 (Vagnon 2020), while the empiri- T through Eq. 10, T can be achieved in Eq. 11 by fixing the 2 1 cal static coefficient κ ranges from 9 to 11 as reported ratio τ = T /T (for example from experimental evidence). 1 2 by Armanini (1997) or in the range 3–30 as observed The description of the impact dynamics is complete. by Scheidl et al. (2013) for Fr < 3 . The static coeffi - cient κ is suggested to be assumed equal to 1 (Ng et al. ∫ F (t)dt = F + F − τ F T 2 (9) peak peak,stat peak,stat 2021) for saturated flows, which are fluidized due to the excess pore pressure developed inside the landslide at the impact. In this paper, the value of α is calibrated based on T = 2mv / F + F − τF 2 0 (10) peak peak,stat peak,stat the MPM simulation of a selected set of realistic cases. The landslide kinetic energy during the impact pro - T = τT 1 2 (11) cess is derived from its velocity variation over time until the impact process finishes ( T ). The impulse theorem Summing up, the primary unknown of such LSI model (Eq. 4), where the impulse of the impact force is equal is T , while the quantities α , κ and τ , must be calibrated. to the variation of linear momentum, the link between Some examples are shown in Fig. 3 to highlight the the impact force and velocity variation is obtained. effect of α , κ and τ on impact force and kinetic energy Since the time-trend of the impact pressure is a piece- trend over time. The input quantities of the model are: wise function, the equations system reads as in Eq. 5. L = 15m; h = 3m; ρ = 1800kg/m ; v = 10m/s . High 1 m 0 For sake of simplicity, it is assumed that t = t = 0. values of α result in large peak forces, short impact time 0 imp T and rapid decrease of the kinetic energy of the flow. This means that α can be interpreted as a measure of I = ∫ F t dt = ∫ mdv ( ) (4) 0 0 system deformability, since the decreasing of T with α means that the system is stiffer. The empirical coefficient κ has similar behaviour com- F t/T 0 < t < T peak 1 1 pared to α , since high values of κ result in large peak forces F [1 − (F /F ) F(t) = . peak peak,dyn peak and short time T . However, the coefficient κ has a minor (t − T )/(T − T )] T < t < T 1 2 1 1 2 2 influence on the system response compared to the coeffi - (5) cient α (as evident in Fig. 3) and its determination is quite The reduction in landslide velocity is obtained from complicated. For this reason, the static component of the Eq. 6, by solving the integrals in Eq. 5 and replacing the impact force could be disregarded ( κ = 0 ), using only the term F (t) with Eq. 4. Thus, the flow velocity over time coefficient α for the assessment of the impact scenario. (Eq. 7) and the corresponding kinetic energy (Eq. 8) can For fluidized flows the assumption of κ = 1 is preferable be computed. (as suggested by Ng et al. 2021) therefore also this value peak 2 will be employed for the calibration of the model. t 0 < t < T 1 2mT 1 � � Finally, the ratio τ governs the occurrence of the peak F F T peak peak,dyn 1 �v(t) = + t (6) m 2m(T −T ) 2 1 time, and thus the shape of the impact force trend. In peak,dyn 2 terms of flow kinetic energy dissipation, the higher the − t T < t < T 1 2 2m(T −T ) 2 1 ratio τ , the steeper the dissipation trend up to T and the slower the energy reduction between T and T . In a 1 2 v(t) = v − �v(t) 0 (7) sense, the parameter τ can be interpreted as a measure of the impulsiveness of the impact loading. Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 5 of 17 landslide is meant to represent the shape of the flow at a certain time during propagation (Fig. 3), but it strongly depends on site-specific flow-path topography and geo - morphological conditions. However, friction with the ground topography often result in a stronger higher front and weaker lower body and a tail of the mixture flow (Iverson 1997; Pudasaini and Fischer 2020; Thouret et al. 2020). Here, we consider a 45°-inclined front and a tail of length equal to three times the flow height. To consider different flow volumes, a i number of squares are placed between the head and tail portions. Given this shape, the landslide as the same volume of an equivalent rectangular with the same height h and a length L = (2 + i) · h , and unitary width (Fig. 3). The flow is a saturated mixture with hydrostatic distribution of initial pore-water pressure. The landslide is assumed as approaching the barrier with a fixed geometric configuration and constant velocity, until the LSI starts. For the sake of simplicity, the flow basal frictional force F (Fig. 1a) is assumed equal to zero in all cases, by means of a smooth contact. Although simpli- fied, the landslide scheme resembles its main character - Fig. 3 Dependence of the impact force (a) and landslide kinetic istics such as velocity, impact height, non-zero interstitial energy (b) on model parameters pressures and elasto-plastic behavior. The mechanical parameters of the landslide material are: ρ (density of the mixture) = 1800 kg/m ; n (porosity) = 0.4; φ’ (eeff c - Calibration via MPM modelling tive friction angle) = 20°; c’ (effective cohesion) = 0; E’ Input and data (Young modulus) = 2 MPa; ν (Poisson’s ratio) = 0.25; k sat −4 The Material Point Method is an appropriate modelling (hydraulic conductivity) = 10 m/s; μ (liquid viscos- −6 alternative for large deformation problems. The Lagran - ity) = 10 kPa s; K (liquid bulk modulus) = 30 MPa. gian points (named Material Points) are free to move across The barrier is modelled through the one-phase single- a fixed mesh, which schematizes the domain where the point MPM formulation (“Appendix 1”). For the barrier materials are in their initial configuration and where they it is assumed: non-porous material, base fixed to the will move during the deformation process. At each time ground and rigid behaviour. This last hypothesis relates step, the governing equations are solved on the mesh, but to the construction mode typically used for such barriers then all the stress–strain variables are saved in the MPs. (Cuomo et al. 2020). To schematize the LSI problem in a realistic way, the Different impact scenarios are investigated. The geo - build-up of pore water pressure in the flow material metric features of both the landslide and the barrier are during the impact is considered as well as the hydro- summarized in Table 1. It is worth noting that the case of mechanical coupled behaviour and the yielding of the an infinite wall is that considered in the literature empiri - flow material. For a saturated porous material, each MP cal models. The expected impact mechanism (Table 1) is reproduces a volume of the mixture V , given by the sum computed for each scenario, based on the diagram pro- of the solid V and liquid V phases volumes. Each MP posed by Faug (2015), with the Froude number (defined S L stores the information about both the solid and liquid as v/ gh ,) calculated considering that the impacted side phases. This is called two-phase single-point formulation of the barrier is inclined of β (i.e., v = v sin β ). From that, (Jassim et al. 2013; Ceccato et al. 2018; Fern et al. 2019). the expected amount of overtopping is inferred. The primary unknowns are the solid ( a ) and the liquid Examples of the geometric schematization (fixed acceleration ( a ). From there, the velocity of solid and background mesh) in the MPM model are provided liquid phases are obtained. The MPs are moved with the in Fig. 4. The computational fixed mesh is always kinematics of the solid skeleton during the computation. unstructured, namely made of triangles with differ- Details are reported in the “Appendix 1” and by Mar- ent sizes, and finer in the zone where the LSI occurs. tinelli and Galavi (2022). For instance, in Fig. 4a, it is made of 20,515 triangular The landslide is modelled through the two-phase sin- 3-noded elements with dimensions ranging from 0.20 gle-point formulation. The initial configuration of the to 1.00 m. On the other hand, the number of MPs is Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 6 of 17 Table 1 Selected impact scenarios ID L L i ( −) h (m) V ν β (°) d (m) L B (m) b (m) H (m) Fr H/h Impact Expected 1 m 1 0,1 2 (m) (m) (m /m) (m/s) (m) ( −) ( −) mechanism* overtopping^ 1 21 15 3 3 45 10 60 3 6.95 11 4 6 1.60 2 Standing jump Large 2 21 15 3 3 45 20 60 3 6.95 11 4 6 3.19 2 Airborne jet Large 3 21 15 3 3 45 10 90 3 ∞ – – ∞ 1.84 ∞ N/D Nil 4 47 45 43 1 45 10 60 3 6.95 11 4 6 2.76 6 Bore Moderate 5 21 15 3 3 45 5 60 3 6.95 11 4 6 0.80 2 Dead zone Nil 6 21 15 3 3 45 15 60 3 6.95 11 4 6 2.40 2 Airborne jet Large 7 21 15 5 3 63 10 60 3 6.95 11 4 6 1.60 2 Standing jump Large 8 21 15 3 3 45 10 80 3 6.08 8.5 6.5 6 1.82 2 Standing jump Large 9 21 15 1 4 48 10 60 3 6.95 11 4 6 1.38 2 Standing jump Large From the study of Faug (2015); ^ from the conceptual scheme in Fig. 1 9535 for case 3 (Fig. 4b), while equal to 24,107 for case Figure 5c reports the velocity distribution of the case 2 4 (Fig. 4f ). with a huge initial kinetic energy of the flow. The barrier cannot stop the propagation of the landslide and a very prolonged jet with high energy is formed after the impact Examples of MPM results thus the amount of material that is retained by the bar- The numerical MPM analyses evidently allow the simul - rier is quite smaller than the standing jump cases (large taneous simulation of flow propagation and flow-struc - overtopping). Interesting is also the case of a shallow ture interaction. One of the main advantages of the MPM flow (case 4 in Fig. 5d), where the flow hits the barrier modelling is the possibility of monitoring important and then withdraws in unsteady conditions (bore impact quantities of the flow during impact, such as stresses, mechanism). The flow does not have enough energy to strains, pore-water pressure, velocity, depth, etc. For overtop the barrier and therefore falls downward creating example, in Fig. 5 is shown the spatio-temporal evolu- some turbulence in the remaining part of the incoming tion of landslide velocity field. The cases 1, 2, 4 and 5 of flow (t > 3 s). Table 1 are chosen since they are characterized by dif- In Fig. 6 the Landslide-Structure-Interaction dynamic ferent impact mechanisms and consequently by diverse process is explained well. The peak of the horizontal and amount of overtopping expected. The numerical simu - vertical components of the impact force ( F and F , 2,x 2,y lation of these cases can validate the conceptual scheme respectively in Fig. 4a and b) are quite different in all reported in Fig. 1, as it is possible to compute the per- the cases considered. In particular, a clear peak force is centage of volume retained by the barrier (V /V ), f,LH 1 attained for the cases 2, 3 and 6. The higher is the flow namely the left hand accumulated landslide volume velocity (or the steeper the barrier), the higher is the (V ) divided to the landslide volume (V ). This is an f,LH 1 peak force. Conversely, the cases 4 and 5 does not show important issue that must be considered for the design of any distinct peak, where the impact forces are very lim- the protection barriers. ited. The frictional force along the top of the barrier ( F ), Specifically, it emerges that for case 1 (Fig. 5a) the which is caused by landslide overtopping, is also illus- impact mechanism is a standing jump with large over- trated in Fig. 6c. The highest F is computed in the case topping. Here, a part of the incoming flow overtops the 2, where the retained volume (V ) is the smallest. This f,LH barrier, forming a jet with high kinetic energy. Moreover, force can also have a negative sign when the flow goes during the formation of the jet, the velocity of the flow upstream, instead of flowing beyond the barrier. upstream of the barrier is almost zero (t = 3 s). Then, the The kinetic energy of the incoming flow ( E ) is plot- flow has lost most of the initial kinetic energy and there - ted in Fig. 7. For a more comprehensive comparison, the fore goes back, also due to the presence of a smooth con- curves are normalized by the initial kinetic energy of the tact along the ground base. Completely different is case 5 flow ( E ). All the curves (except for case 4) show a sud- k,0 (Fig. 5b), where the flow is characterized by a lower initial den reduction, reaching the minimum value at approxi- velocity. The impact mechanism here is the formation of mately t 2s and, after that, the energy increases again as a dead zone and all the flow is completely blocked by the the formed jet takes the downward direction. This means barrier. that, during the flow, the kinetic energy is transformed to Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 7 of 17 Fig. 4 Schematization of the problem in the numerical MPM model: a cases 1, 2, 5, 6; b case 3; c case 7; d case 8; e case 9; f case 4 potential energy. For t > 2s all the trends are very differ - Calibration results ent, since the curves represent the kinetic energy of the The calibration of the empirical model principally focuses overcoming jet (especially for case 2) combined with the on the evaluation of the coefficients τ and α . The parame - energy of the reflecting flow (especially for cases 3 and ter τ is obtained by imposing the equivalence between T 5). The behavior of case 4 is completely different: it is calculated from Eq. 11, and T obtained from the MPM characterized by a slower and constant reduction of the simulations. The coefficient α relates to relevant features energy as the impact mechanism does not induce the for- of the flow such as the grain size distribution, the barrier mation of any jet. type and the flow-structure interaction mechanism such Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 8 of 17 as the formation of vertical jet-like wave at the impact The output of the proposed empirical method are com - (Canelli et al. 2012). As reported in the literature, this pared with the numerical results as it concerns the most parameter can vary in a wide range (between 0.4 and relevant factors in LSI for the case with κ = 0 (Fig. 9). A 12), often leading to an excessive overestimation of the similar comparison for the case with κ = 1 was also per- design impact load. However, many authors (Hübl et al. formed with satisfactory results, and it is omitted here for 2009; Proske et al. 2011; Scheidl et al. 2013; Cui et al. the sake of the simplicity. Based on the above calibration, 2015; Vagnon 2020) developed a power law relationship the values of F and T computed through the empiri- peak 1 between the coefficient α and the Froude number ( Fr ), a s cal method (Eqs. 2 and 11, respectively) fit very well the reported in Eq. 12. MPM numerical results for all the scenarios. On the other hand, it is observed that the impact period T (com- α = a Fr (12) puted from Eq. 10) is only slightly overestimated by the empirical method especially for those cases with higher The evaluation of the coefficients a and a requires at 1 2 velocities. In these cases, the empirical method is not least two numerical simulations with different Froude able to consider the amount of material which overtops number. All the cases of Table 1 (except for cases 4 and 5, the barrier. In fact, as the mass m decreases, this mate- neglected due to the impossibility of identifying a unique rial no longer contributes to the variation of the linear peak value) are used for the calibration of the model. The momentum of the landslide (Eq. 4), therefore a lower case 1b is added with a different soil porosity. Globally, value of T is expected from Eq. 10. Only if the empiri- the influence of soil porosity n, landslide thickness h , cal equation is applied to the condition of the indefinite landslide volume V , initial velocity v and the barrier 1 1,0 wall (case 3), where the overtopping of the barrier is not side inclination β are considered (Table 2). allowed, then the empirically-computed time T perfectly Two calibration procedures were followed. The first matches the MPM outcome. In this case, even the other neglects the static component of the impact force, thus calculated quantities correspond to those obtained from the model can be considered purely hydro-dynamic. MPM since the indefinite wall most resemble the basic The other one assumes an empirical static coefficient κ assumptions of the empirical model. equal to 1, that is more plausible for saturated flows. In For the evaluation of the flow kinetic energy at the peak the latter case, the peak impact force resulting from the impact force time, i.e., E (T ), the empirical formula- k 1 MPM simulations was depurated of the static component tion provides lower values than MPM for the cases with ( 0.5ρ gh ) for obtaining the dynamic one. The best fit v > 10m/s , while there is an appreciable matching for values are a = 1.781 and a =−0.515 for κ = 0 (Eq. 13) 1 2 the other cases. This is mainly caused by the inability of and a = 1.432 and a =−0.365 for κ = 1 (Eq. 14). The 1 2 the simplified proposed method to consider the hydro- calibrated value for τ is 0.14 for all the cases. The results mechanical coupling and large deformations within the show a good fitting with the α − Fr curve for all the flow, which play a crucial role during the interaction with impact scenarios, and it is relevant that also the trend the obstacle. over time of the impact force is reproduced quite faith- fully for both. κ = 0 (Fig. 8a) and κ = 1 (Fig. 8c). Validation for a large dataset −0.515 2 F = 1.781Fr ρ v h peak m (13) 1,0 The proposed empirical method is thoroughly validated towards the interpretation of a large dataset of real obser- −0.365 2 2 F = 1.432Fr ρ v h + 0.5ρ gh (14) vations of flow-type landslides, achieved through a per - peak m m 1,0 manent monitoring station. The field dataset from Hong Besides achieving a good correspondence with the et al. (2015) includes thickness, density, channel width, impact forces, the trend of flow kinetic energy was com - volume of discharge, velocity and impact forces recorded puted (Eq. 8) for the impact scenarios (Fig. 8b and d), in real time during debris flow events. giving for instance better agreement for the case 1 than The data are relative to 139 historical events that took the case 2. In the latter case, this is explained by the fact place between 1961 and 2000 in the Jiangjia Ravine that high energy of the flow produces a more elongated basin, located in the Dongchuan area of Yunnan Prov- jet, which cannot be reproduced by a simplified empirical ince in China (Zhang and Xiong 1997; Kang et al. 2007; method. The flow kinetic energy computed via empiri - Hong et al. 2015). The bulk density ranges from 1600 cal method is always lower than that computed through to 2300 kg/m with fluid concentration ranging from MPM (apart from case 7). This is mostly linked to the 0.15 to 0.6. The dataset is well suited for the valida - simplifying hypothesis of neglecting the static compo- tion purpose as wide ranges of the relevant features are nent of the impact force. However, it is a safe approxima- tion, to be considered acceptable in the practice. Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 9 of 17 Fig. 5 Velocity field at different time lapses: a case 1 (standing jump); b case 5 (dead zone); c case 2 (airborne jet); d case 4 (bore) Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 10 of 17 Fig. 6 Impact forces for different scenarios: a horizontal component of F ; b vertical component of F , c F 2 2 3 considered such as: v = 3 − 14m/s , h = 0.2 − 2.7m , MPM is an advanced numerical method and has 6 3 V = 269 − 1.75 · 10 m and p = 14 − 435kPa. proved to be reliable in predicting the impact force peak The impact peak pressure is calculated through the trend over time (Cuomo et al. 2021). Moreover, unlike calibrated power law for the peak force (Eq. 14) as fol- field evidence or laboratory tests, the numerical results −0.365 2 lows: p = 1.432Fr ρ v + ρ gh.The results are provide additional features, through the computation m m peak 1,0 reported in Fig. 10 and show a very good correspondence and time–space tracking of different quantities, such with the field data, being the difference much less than as stress, strain, pore pressure, solid and liquid veloci- 10% for most of the cases. In particular, the empirical ties, which cannot be easily monitored or obtained in model predicts quite well the peak of impact pressure for the field. Particularly focusing on LSI, many advantages low values but showing some dispersion for values higher come from using MPM. Primarily, it allows considering than 150 kPa. The statistical distribution of the error, all such important features of the saturated flows, i.e. obtained as the difference between the computed value hydro-mechanical coupling and large deformations dur- and the measured value, shows that the median value is ing propagation and impact. The accurate knowledge of 10.7 kPa and the 90th percentile value is 37.65 kPa. The the impact mechanism and so the evolution of flow depth application of the numerical MPM model to such a large and velocity is crucial for the design of mitigation coun- field dataset is beyond the scope this paper, while it could termeasures. For example, the accurate estimate of the be a future development. length of the vertical jet must prevent that the retaining structure is overtopped by the flow, thus being ineffec - Discussion tive. However, MPM suffers from some limitations, such A comparison between the presented methods is neces- as the high computational cost and until now the diffi - sary to assess their strengths and weaknesses for analyz- culty of being available in engineering practice. ing the Landslide-Structure-Interaction. Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 11 of 17 Fig. 7 Kinetic energy of the flow for different scenarios Table 2 Selected parameters for the calibration of the empirical variability of the achieved results. The empirical model model through MPM simulations proposed in the present paper has the highest corre- spondence among the real data and the computed values, Flow type landslide Barrier with a contained dispersion of the results. Some discrep- ID n ( −) h (m) V (m /m) ν (m/s) β (°) 1 1,0 ancy of the results for very high velocities. Can the present method be applied to the real multi- 1 0.5 3 45 (i = 3) 10 60 phase debris flows? The answer is positive, especially in 1b 0.3 3 45 (i = 3) 10 60 those cases where there are options on where to build one 2 0.5 3 45 (i = 3) 20 60 or more of these barriers. Hence, a quick direct method 3 0.5 3 45 (i = 3) 10 90 to assess the actions transferred by the flow to one bar - 6 0.5 3 45 (i = 3) 15 60 rier is helpful in the preparation of the whole mitigation 7 0.5 3 63 (i = 5) 10 60 plan. However, the LSI problem is not only a matter of 8 0.5 3 45 (i = 3) 10 80 peak impact pressure or kinetic energy decay. In fact, 9 0.5 4 48 (i = 1) 10 60 any barrier may suffer of flow overtopping, deformation/ damage and shifting under the impact (and during the LSI). This is the reason that such flows and flow struc - Empirical methods are more immediate and easier ture interactions are being simulated with the advanced to use than MPM, since they provide an estimate of multi-phase mass flow models such the MPM approach the impact quantities considering only the flow den - proposed here or through other advanced models (Mer- sity, thickness and velocity as input and thus they could gili et al. 2020). be preferable in the assessment of the LSI problems for design purposes. Concluding remarks Here, we apply some empirical methods available in the The present paper has proposed a conceptual framework, literature to interpret the field data of Hong et al. (2015), empirical and numerical models to analyse the impact used already in Sect. 5, and we also applied the pro- of flow-like landslides against artificial barriers, focusing posed method to the same dataset. The chosen empiri - not only on the evaluation of the peak impact forces but cal formulations are those of Hübl and Holzinger (2003), also on the kinematics of the landslide during the whole Armanini et al. (2011), Cui et al. (2015) and Vagnon impact process. A conceptual framework for the Land- (2020), all classifiable as mixed models (refer to Sect. 1). slide-Structure-Interaction (LSI) problem has been firstly In terms of peak pressure, the results of the empirical introduced to better focus the main variables that govern models of Armanini et al. (2011) and Vagnon (2020) have the dynamics of the impact process. This framework has a low dispersion in the plot of Fig. 11, but with an overes- been then implemented in a novel empirical method. timation of 61% and 35%, respectively. The formulations The calibration of the new proposed empirical method proposed by Hübl and Holzinger (2003) and Cui et al. has been performed using a set of numerical analyses (2015) are, in contrast, characterized by a quite relevant Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 12 of 17 Fig. 8 Calibration of the empirical model through the MPM simulations (for the cases in Table 2): impact force (a) and kinetic energy (b) time trend for κ = 0 and impact force (c) and kinetic energy (d) time trend for κ = 1 conducted through the Material Point Method (MPM). of peak pressures for values larger than 350 kPa, so it The latter can easily reproduce a wide range of impact must be used with caution. Nevertheless, some available scenarios considering all the main features of LSI, such literature methods have been also applied to the same as the hydro-mechanical coupling, the soil large defor- database, and thus the advantages of the new method are mations and the presence of multi-materials. It has been outlined. possible to derive a new α − Fr power law relationship In conclusion, both the proposed numerical and to derive the peak impact pressure. This formulation is empirical methods can appropriately simulate the physi- different from those in the literature, which are typically cal phenomena. Particularly, the numerical MPM analy- calibrated on small-scale laboratory tests, thus giving an ses evidently allow the simultaneous simulation of flow excessive overestimation in predicting the impact load propagation and flow-structure interaction. On the that may results in a large increment of costs for struc- other hand, generally good agreement between empiri- ture construction. cal model and MPM simulation indicates that both The validation of the empirical method has been done results are physically meaningful. Further research may referring to a vast dataset of real field evidence col - be directed to an enhancement of the proposed empiri- lected at Jiangjia Ravine (China). The achieved results are cal model considering the amount of material that may encouraging, showing a high correspondence between overtop the barrier, giving more accurate results for the the output of the proposed empirical formulation and the analysis of the LSI problem. measured field data. However, the estimated power law for the empirical model can lead to an underestimation Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 13 of 17 Fig. 9 Comparison of proposed empirical model with MPM results (κ = 0) liquid phases. The accelerations of the two phases are the primary unknowns: the solid acceleration a , which is calculated from the dynamic momentum balance of the solid phase (Eq. 15), and the liquid acceleration a , which is obtained by solving the dynamic momentum balance of the liquid phase (Eq. 16). The interaction force between solid and liquid phases is governed by Darcy’s law (Eq. 17). Numerically, these equations are solved at grid nodes considering the Galerkin method (Luo et al. 2008) with standard nodal shape functions and their solutions are used to update the MPs velocities and momentum of each phase. The strain rate ε ˙ of MPs is computed from the nodal velocities obtained from the nodal momentum. n ρ a =∇ · (σ − np I) + (ρ − nρ )b + f S S S L m L d (15) Fig. 10 Application of the proposed empirical model to the large ρ a =∇p − f L L L d (16) field dataset (139 cases) collected by Hong et al. (2015) nµ f = (v − v ) L S (17) Appendix 1: Material point method model The resolution of solid and liquid constitutive laws equations (Eqs. 18 and 19) allows calculating the increment of effec - The equations to be solved concern the balance of tive stress dσ and excess pore pressure dp , respectively . dynamic momentum of solid and liquid phases, the mass The mass balance equation of the solid skeleton is then balances, and the constitutive relationships of solid and Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 14 of 17 used to update the porosity of each MP (Eq. 20), while used to mitigate volumetric locking is the strain smooth- the total mass balance serves to compute the volumetric ening technique, which consists of smoothing the volu- strain rate of the liquid phase (Eq. 21) since fluxes due to metric strains over neighbouring cells. The reader can spatial variations of liquid mass are neglected (∇ nρ = 0). refer to Al-Kafaji (2013) for a detailed description. Regarding the critical time step, the influence of per - dσ = D · dε (18) meability and liquid bulk modulus must be considered as well (Mieremet et al. 2016). In particular, the time step dp = K · dε L L vol (19) required for numerical stability is smaller in soil with lower permeability (Eq. 22). Dn = n ∇· v = 0 (20) d S S Dt �t = min √ ; cr (E + K /n)/ρ L m (22) 2(ρ + (1/n − 2)ρ )k m L sat Dε n vol S = ∇· v + ∇ · v (21) S L ρ g Dt n The sliding modelling of the flowing mass on the rigid In the two-phase single-point formulation the liquid material is handled by a frictional Mohr–Coulomb mass, and consequently the mass of the mixture, is not strength criterion. The contact formulation was used to constant in each material point but can vary depend- ensure that no interpenetration occurs, and the tangen- ing on porosity changes. Fluxes due to spatial variations tial forces are compatible with the shear strength along of liquid mass are neglected and Darcy’s law is used to the contact. The reaction force acting on the structure at model solid–liquid interaction forces. For this reason, node j was calculated as in Eq. 23. this formulation is generally used in problems with small gradients of porosity, and laminar and stationary flow in F (t) = m �a + m �a j j,S S,contact j,L L,contact (23) slow velocity regime. However, this formulation proves to be suitable for studying flow-structured-interaction The terms a and a are the change S,contact L,contact (Cuomo et al. 2021). The water is assumed linearly com - in acceleration induced by the contact formulation, for pressible via the bulk modulus of the fluid K and shear both solid and liquid phase, and m and m are the cor- i,S i,L stresses in the liquid phase are neglected. responding nodal masses. The total reaction force is the The current MPM code uses 3-node elements which integral of the nodal reaction forces along the barrier. suffer kinematic locking, which consists in the build-up of fictitious stiffness due to the inability to reproduce the Appendix 2: A note on the role of pore water correct deformation field (Mast et al. 2012). A technique pressure Selected results are shown in Figs. 12 and 13, where the spatial distribution of pore-water pressure is illustrated at different time instants of the propaga- tion stage for all scenarios of Table 1. During the im- pact, the initial liquid pressure (< 30kPa ) changes over time, with the maximum value in the first instants of the impact process ( t = 1s ) and later diminishing down to nil in some cases. However, the maximum value of pore water pressure ( p ) is dependent on the type of bar- L,max rier. In fact, comparing an infinite wall (Fig. 12a) with a fixed artificial barrier (Fig. 12b), it follows that p is L,max higher in the first case, where the overtopping is impos - sible, and the impacted area of the barrier is larger than for the artificial barrier ( t = 1s ). At t = 2s , the flow overtops the wall (Fig. 12a) or goes beyond the barrier forming a prolonged jet (Fig. 12b). Liquid pressure is Fig. 11 The proposed empirical method compared to the results decreasing, indicating that we are in the decay zone of of some literature empirical methods in the interpretation of field the impact force diagram. Subsequently ( 4s < t < 6s ), dataset (139 cases) collected by Hong et al. (2015) the flow loses more and more energy and falls down- Di P erna et al. Geoenvironmental Disasters (2022) 9:8 Page 15 of 17 Fig. 12 Pore-water pressure distribution for: a infinite wall (case 3); b fixed barrier (case 1) Fig. 13 Pore-water pressure distribution for different flows: a v = 20 m/s (case 2); b H/h = 6 (case 4) 0 Di Perna et al. Geoenvironmental Disasters (2022) 9:8 Page 16 of 17 3 3 Poisson’s ratio; : Liquid density; : Density of the mixture; wards (similarly, in both cases). ρ kg/m ρ kg/m L m : Solid density; : Total stress tensor of the mixture; ρ kg/m σ (kPa) σ ˙ (kPa/s) The expected impact mechanism, as assessed from ′ ◦ : Jaumann stress rate matrix; : Normal stress; : Internal friction σ (kPa) ϕ ( ) the use of the diagram by Faug (2015), is confirmed in angle; : Dilatancy angle. ψ( ) both the cases. For the infinite vertical wall, the impact Acknowledgements mechanism resembles the bores regime since a granular The research was developed within the framework of Industrial Partnership jump (named “bore”) is formed which heads upstream Ph.D. Course (POR Campania FSE 2014/2020). All the MPM simulations were performed using a version of Anura3D developed by Deltares. of the wall. For the embankment barrier, the impact mechanism is the standing jump, which is similar to Author contributions the bore regime but here a part of the incoming flow is ADP is responsible for conceptualization, data collection and numerical mod- elling. SC is responsible for the data collection, numerical modelling and the able to overtop the barrier, forming a jet with very low corresponding passages in the manuscript. MM is responsible for numerical energy. modelling and conceptualization. The authors read and approved the final Overall, the cases 1, 2 and 4 suggest a clear link manuscript. between pore-water pressures at impact and the Funding amount of overtopping flow mass, where larger pore- The research was supported by several Italian Research Projects funded by the water pressures facilitate the overtopping of the barrier. Italian Education and Research Ministry such as: Project FARB 2017 “Numeri- cal modelling and inverse analysis for flow-like landslides”; Project FARB 2014 This finding is also confirmed by previous experimental “Large area analysis of triggering and propagation landslide susceptibility for research (Song et al. 2017; Zhou et al. 2018). flow-like landslides”; Project FARB 2012 “New Frontiers of advanced numerical Different flows are also considered to outline how simulation of destructive landslides”. MPM can reproduce the various impact mechanisms (Fig. 13). High flow velocities induce large values of Declarations p , which reaches 260 kPa (Fig. 13a). In this case, L,max Competing interests the expected impact mechanism is an airborne jet The authors declare that we have any competing financial interests. (Table 1) and it is confirmed very well from the numeri - Author details cal simulation. A very prolonged jet with high energy is 1 2 3 University of Salerno, Fisciano, Italy. Deltares, Delft, Netherlands. T echnical formed after the impact, thus the amount of material University of Delft, Delft, Netherlands. that is retained by the barrier is much smaller than the Received: 23 December 2021 Accepted: 3 April 2022 standing jump case. Completely different is the case of a shallow flow (Fig. 13b), where the flow hits the obsta - cle and propagates upstream in unsteady conditions (bore regime). References Al-Kafaji I (2013) Formulation of a dynamic material point method (MPM) for geomechanical problems. 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Geoenvironmental Disasters – Springer Journals
Published: Apr 15, 2022
Keywords: Impact mechanisms; Mitigation; Structure; Material Point Method
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