A variant of finite element method: Deformable

Report ARA: 5 September 20, 2018, Alek & Research Associates, LLC and University of New Mexico

A variant of finite element method: Deformable discrete element method Aleksander Zubelewicz Alek and Research Associates, LLC, Los Alamos, NM 87544 University of New Mexico, Civil Engineering, Albuquerque, NM 87106 [email protected] MOTIVATION

DOI:

It is worth to remember that the first discrete element methods were formulated about five decades ago. This report summarizes a long forgotten description of the deformable discrete element method (DDEM). The method was proposed in early 70s. Unfortunately, only simplified versions of the concept were reported, while the original version of the method was forgotten. I strongly believe that computational mechanics community should be interested learn about the original DDEM version. A complete description of the method was reported in my Ph.D. thesis. Even though written in Polish, relevant equations and figures clearly confirm foundation of the method. For completeness, the thesis is made a part of my report. Also, I included a novel constitutive model for brittle materials.

ABSTRACT The deformable discrete element method (DDEM) was first reported in 1975. I believe that this might be one of the first, if not the first, DEM approach used for the prediction of fracture processes in otherwise continuous brittle materials. Later, the method was reduced to a system consisting of rigid elements and springs, while the description of deformable particles was forgotten. In here, I describe the method as originally developed. Each particle is subjected to translations and rotations, while particle deformation is described with the use of shape functions. In a general case, particle rotations act on moments, and then, the system resembles a pseudo-Cosserat medium. In the case of unconstrained rotations, the method properly approximates classic continuum solutions. In this approach, mass and moments of inertia are placed in the particle mass centers. A portion of mass is also lumped into external nodes of each particle. As a result, equations of motion are developed in internal nodes (mass centers), while external nodes connect discrete elements. I show capabilities of the method for brittle materials subjected to dynamic loading. KEYWORDS: Discrete element method, Deformable particles, Constitutive equations, Brittle materials, Fracture

springs7, where the spring breakage constituted particle-to-particle cracking.

1. INTRODUCTION The deformable discrete element method is believed to be one of the first, if not the first, discrete element method designed for the prediction of fracture processes in brittle materials1, 2. Simplified versions of the method were designed to study the behavior and failure of concrete3, 4, rock burst processes in deep mines5 and, lastly, sitting mechanisms of weathered rocks subjected to dynamic excitations6. In these implementations, the rigid cores were interconnected by a system of

In the same timeframe, rigid element (block) methods were used in the studies of static and dynamic behavior of plates and beams8, were found beneficial in an analysis of ship structure dynamics9, properly reproduced seismic responses of structures10, were applied to study large scale movement of loose blocks of rock11, and correctly replicated flow processes of granular materials12. Since then, the discrete element methodology became an established branch of computational

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mechanics, thousand papers were written, books were published and careers were made.

Local rotations may exert additional strain

I emphasize that this report is not a review the DEM advances. My intent is to communicate the forgotten description of the deformable discrete element method. Since 70th, many features of the method have been accounted for in various numerical codes. Still, I believe that the DDEM structure is unique and worth reporting. Recently, a simplified version of the concept has been published13, but the method as originally developed remains in the shadows.

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𝒇𝒇𝑖𝑖𝑖𝑖 = ∫�𝑩𝑩𝑇𝑇 ∙ 𝝈𝝈 + 𝑩𝑩𝑇𝑇𝜑𝜑 𝜇𝜇 𝛽𝛽𝜑𝜑 𝜀𝜀𝜑𝜑𝑖𝑖 � 𝑑𝑑𝑉𝑉𝑠𝑠𝑖𝑖 .

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𝑖𝑖 ⁄ 𝑟𝑟𝑖𝑖 �⁄𝑁𝑁𝑖𝑖 . In here, where we have 𝜀𝜀𝜑𝜑𝑖𝑖 = �∑𝑁𝑁𝑖𝑖 Δ𝑢𝑢𝜑𝜑 𝑖𝑖 relative displacements Δ𝑢𝑢𝜑𝜑 result from rigid rotations only. Vectors 𝒓𝒓𝑖𝑖 (length 𝑟𝑟𝑖𝑖 ) connect the appropriate external node with the mass center. The rotation-induced strain is responsible for additional 2 storage of elastic energy 𝑈𝑈 𝜑𝜑 = 𝜇𝜇 𝛽𝛽𝜑𝜑 �𝜀𝜀𝜑𝜑𝑖𝑖 � ⁄2 and plays important role during dynamic behavior of heterogeneous materials. Shear modulus is 𝜇𝜇 and the parameter 𝛽𝛽𝜑𝜑 scales the rotation-induced constraints. Nodal forces in external joints are

2. DEFORMABLE DISCRETE ELEMENTS Each discrete element consists of a rigid core surrounded by a deformable boundaries, Fig. 1.

Stress 𝝈𝝈 is calculated in each particle 𝝈𝝈 = 𝑪𝑪 ∙ 𝜺𝜺𝑒𝑒𝑖𝑖𝑖𝑖 , where elastic strain is 𝜺𝜺𝑒𝑒𝑖𝑖𝑖𝑖 and elastic matrix is denoted as 𝑪𝑪. Equations of motion are constructed in external nodes and in mass centers (internal nodes) 𝑭𝑭𝑒𝑒 = ∑𝑁𝑁𝑖𝑖 𝒇𝒇𝑒𝑒𝑒𝑒 + 𝑀𝑀𝑒𝑒 𝒖𝒖̈ 𝑒𝑒 𝑭𝑭𝑖𝑖 = ∑𝑁𝑁𝑖𝑖𝑖𝑖 𝒇𝒇𝑖𝑖𝑖𝑖 + 𝑀𝑀𝑖𝑖 𝒖𝒖̈ 𝑖𝑖 . 𝑴𝑴𝜑𝜑 = ∑𝑁𝑁𝑒𝑒 𝒓𝒓𝑖𝑖 × 𝒇𝒇𝑖𝑖𝑖𝑖 + 𝐽𝐽𝑀𝑀 𝝋𝝋̈𝑖𝑖

(4)

Particles interactions are monitored in external nodes 𝑁𝑁𝑖𝑖 . Of course, the number of interconnected particles varies from node to node and can be changed during damage processes. Forces in mass centers are collected from 𝑁𝑁𝑖𝑖𝑖𝑖 external nodes. Lastly, moments are calculated as described in Eqn. 4 and act on rotational accelerations. In this notation, mass in external nodes is 𝑀𝑀𝑒𝑒 , mass in internal nodes is 𝑀𝑀𝑖𝑖 and moment of inertia becomes 𝐽𝐽𝑀𝑀 . Mass is redistributed between the external and internal nodes such that inertia forces due rotations and translations are mutually decoupled.

Figure 1: Motion and internal deformation of a discrete elements. In past, particle deformation was approximated by springs attached to the core. In the forgotten version of the method, the particle deformation is described with the use of shape functions. Particles (discrete elements) are moving in space and are allowed to experience internal deformation. The motion of particle cores includes translations 𝒖𝒖𝑖𝑖 and rotations 𝝋𝝋 of mass centers (i-nodes), while element-to-element interactions are monitored in eexternal nodes. Nodal relative displacements are measured between points 𝑒𝑒′ and 𝑒𝑒′′, Fig. 1. Consequently, this leads to Δ𝒖𝒖 = 𝒖𝒖𝑒𝑒 − 𝒖𝒖′𝑒𝑒 , and then, the relation becomes Δ𝒖𝒖𝑒𝑒𝑒𝑒 = 𝒖𝒖𝑒𝑒 − (𝒖𝒖𝑖𝑖 + 𝒓𝒓 × 𝝋𝝋). In this expression, vectors 𝒓𝒓 connect selected external node with the mass center. Shape functions redistribute relative displacements inside particles and are fixed in the mass centers. In the next step, the shape functions are converted to deformation matrices 𝑩𝑩 such that strains are expressed in terms of nodal relative displacements 𝜺𝜺𝑖𝑖𝑖𝑖 = 𝑩𝑩 ∙ Δ𝒖𝒖𝑒𝑒𝑒𝑒 .

𝜀𝜀𝜑𝜑𝑖𝑖 = 𝑩𝑩𝝋𝝋 : 𝛥𝛥𝒖𝒖𝑒𝑒𝑒𝑒 ,

3 FRACTURE MODEL FOR CONCRETE In previous applications, DDEM was used to study fracture processes in brittle materials, where cracks propagated along particle interfaces3, 4, 5. In here, I am showing that the method is equally successful in predicting the growth of damage in terms of a continuum-based description. The constitutive equations were previously developed for rocks14 and now are recalibrated for high strength concrete. In the most generic form, the constitutive model can be explained as follows. First, we determine mechanisms of plastic flow. In here, it is the Mohr-

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Coulomb mechanism constructed on the basis of eigentensors 𝑵𝑵1𝜎𝜎 , 𝑵𝑵𝜎𝜎2 and 𝑵𝑵𝜎𝜎3 such that 𝜎𝜎1 = 𝑵𝑵1𝜎𝜎 : 𝝈𝝈, 𝜎𝜎2 = 𝑵𝑵𝜎𝜎2 : 𝝈𝝈 and 𝜎𝜎3 = 𝑵𝑵𝜎𝜎3 : 𝝈𝝈, where 𝜎𝜎1 ≥ 𝜎𝜎2 ≥ 𝜎𝜎3 . The flow tensor is 𝑴𝑴𝑝𝑝 = (1 + 𝑞𝑞𝐷𝐷 ) 𝑵𝑵1𝜎𝜎 − (1 − 𝑞𝑞𝐷𝐷 ) 𝑵𝑵𝜎𝜎3

The model is calibrated for concrete. All parameters are listed in Tables 1 and 2. Selected stress-strain responses include uniaxial compression, stresstriaxiality tests, tension at various strain rates and strain controlled cycles, Fig 2. Stress-strain responses under unconfined and confined compression are shown on the left-hand side of Figure 2. Confining pressure magnifies the material’s strength and decreases strain softening. The responses are known to exhibit strong sensitivity to the size of tested samples. In here, size of the discrete element is small and, for this reason, the softening slope is assumed to be mild.

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and the rate of plastic strain becomes 𝜺𝜺̇ 𝑝𝑝 =

1 2

𝑴𝑴𝑝𝑝 𝑒𝑒̇𝑒𝑒𝑒𝑒 .

(6)

The rate of equivalent plastic strain is coupled with equivalent stress 𝜎𝜎𝑒𝑒𝑒𝑒 = 𝑴𝑴𝑝𝑝 : 𝝈𝝈⁄2 and the relation is 𝑒𝑒̇

𝑒𝑒̇𝑒𝑒𝑒𝑒 = Λ0 𝑒𝑒̇0 � 𝑁𝑁 � 𝑒𝑒̇ 0

𝜔𝜔𝑝𝑝

�

𝜎𝜎𝑒𝑒𝑒𝑒 𝜎𝜎𝑝𝑝

𝑛𝑛𝑝𝑝

� ,

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where Λ0 , 𝜔𝜔𝑝𝑝 and 𝑛𝑛𝑝𝑝 are constants, while 𝑒𝑒̇0 = 1⁄𝑠𝑠 represents the time scale defined here in seconds. Strength 𝜎𝜎𝑝𝑝 is affected by the pre-existing and stress-induced damages. The growth of damage occurs in the orientations determined by the complementary flow tensor 𝑴𝑴𝑐𝑐 = (1 + 𝑞𝑞𝐷𝐷 ) 𝑵𝑵1𝜎𝜎 + (1 − 𝑞𝑞𝐷𝐷 ) 𝑵𝑵𝜎𝜎3 . In this manner, the rate of damage is controlled by the rate 1 𝛀𝛀̇ = 𝑴𝑴𝑐𝑐 𝜉𝜉𝑐𝑐0 2

𝜎𝜎𝑒𝑒𝑒𝑒 𝑒𝑒̇𝑒𝑒𝑒𝑒 𝐺𝐺𝑓𝑓

,

Figure 2: Stress-strain responses are presented for high strength concrete. A sample is subjected to uniaxial compression at strain rate 1/s and confining pressures are 20, 40 and 60MPa. Responses in uniaxial tension are conducted at strain rates 10-3/s, 10-1/s, 10/s and 103/s. Strain-controlled cycles are calculated at strain rate 1/s and within the strain range +/- 0.005.

(8)

and then, the damage tensor becomes 𝛀𝛀 = 𝛀𝛀0 + ∫ 𝛀𝛀̇ 𝑑𝑑𝑑𝑑. The parameter 𝜉𝜉𝑐𝑐0 is equal to one in tension and is much smaller in compression. Note that the complementary and flow tensors have the same squared forms, namely (𝑴𝑴𝑝𝑝 )2 = (𝑴𝑴𝑐𝑐 )2 . We assume that the material may contain pre-existing damages such as shrinkage-induced cracks and other defects 𝛀𝛀0 . We introduce the damage parameter Ω𝐷𝐷 = √𝛀𝛀: 𝛀𝛀, and then, the tensorial representation of the accumulated damage is given by 𝑵𝑵Ω = 𝛀𝛀⁄Ω𝐷𝐷 . Note that 𝑵𝑵Ω : 𝑵𝑵Ω = 𝟏𝟏. The damage growth is proposed in the following form 𝜂𝜂 = 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 𝜉𝜉𝑐𝑐1 𝜂𝜂0 and 𝜂𝜂0 = Ω𝐷𝐷 𝑛𝑛𝑓𝑓 ⁄(1 + Ω𝐷𝐷 𝑛𝑛𝑓𝑓 ). In here, the exponent 𝑛𝑛𝑓𝑓 is a constant. Sign of normal traction 𝜎𝜎Ω = 𝑵𝑵Ω : 𝝈𝝈 determines whether the fracture zone is in compression or tension. In tension (𝜎𝜎Ω ≥ 0), the damage scaling parameter 𝜉𝜉𝑐𝑐1 is equal to one and the parameter is significantly smaller in compression. Also, the damage is responsible for the reduction of strength 𝜎𝜎𝑝𝑝 = 𝜎𝜎𝑝𝑝0 (1 − 𝜂𝜂0 ). Lastly, the damage affects elastic stiffness 𝑪𝑪 = 𝜆𝜆 𝚫𝚫⨂𝚫𝚫 + μ 𝑰𝑰𝐷𝐷 , where the capital delta is 𝚫𝚫 = 𝟏𝟏 − η 𝑵𝑵1Ω and the forth order damage tensor 𝑰𝑰𝐷𝐷 is (𝐼𝐼𝐷𝐷 )𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = Δ𝑖𝑖𝑖𝑖 Δ𝑗𝑗𝑗𝑗 + Δ𝑖𝑖𝑖𝑖 Δ𝑗𝑗𝑗𝑗 .

The viscoplastic constitutive model produces strain rate-dependent responses. At the same time, high strain rates tend to steepen strain softening. Strain controlled cycles show that that the degradation of elastic stiffness is different in tension and compression. The constitutive description includes simplified equation of state. Typically, equation of state is defined in terms of energy and mass density. As we know, concrete is a highly heterogeneous material and, for this reason, dilatational deformation competes with void crushing mechanism. Consequently, the material densification is associated with damages, which affect the wave speed. These microstructural complexities can be avoided by assuming that bulk and shear moduli are directly linked to hydrostatic pressure 𝐵𝐵 = 𝐵𝐵0 [1 + (𝑝𝑝⁄𝑝𝑝𝐸𝐸𝐸𝐸𝐸𝐸 )𝑛𝑛𝐸𝐸𝐸𝐸𝐸𝐸 ] and 𝜇𝜇 = 𝜇𝜇0 [1 + (𝑝𝑝⁄𝑝𝑝𝐸𝐸𝐸𝐸𝐸𝐸 )𝑛𝑛𝐸𝐸𝐸𝐸𝐸𝐸 ], where 𝑝𝑝𝐸𝐸𝐸𝐸𝐸𝐸 and 𝑛𝑛𝐸𝐸𝐸𝐸𝐸𝐸 are constants.

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4 TEST CASES

3 =− 𝐵𝐵11

𝑦𝑦3 −𝑦𝑦𝑖𝑖 𝐷𝐷3

3 𝐵𝐵31 =

4.1 Plane strain: Triangle elements

3 ; 𝐵𝐵15 =

1 𝐵𝐵22

2

𝑦𝑦1 −𝑦𝑦𝑖𝑖

3 ; 𝐵𝐵32 =

𝐷𝐷3

3 𝐵𝐵11

2

3 ; 𝐵𝐵22 = 3 ; 𝐵𝐵35 =

𝑥𝑥3 −𝑥𝑥𝑖𝑖 𝐷𝐷3

3 𝐵𝐵26

2

3 ; 𝐵𝐵26 =−

3 ; 𝐵𝐵36 =

3 𝐵𝐵15

𝑥𝑥1 −𝑥𝑥𝑖𝑖

2

𝐷𝐷3

.

We use triangle elements, where displacements are fixed in the mass center {𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 }, Fig.3. For simplicity, each element consists of three sub-elements. Relative displacements in each kth sub-element are

The denominators are

δ𝑢𝑢𝑘𝑘𝑘𝑘 = 𝐴𝐴𝑘𝑘𝑘𝑘 (𝑥𝑥 − 𝑥𝑥𝑖𝑖 ) + 𝐵𝐵𝑘𝑘𝑘𝑘 (𝑦𝑦 − 𝑦𝑦𝑖𝑖 ) , δ𝑢𝑢𝑘𝑘𝑘𝑘 = 𝐴𝐴𝑘𝑘𝑘𝑘 (𝑥𝑥 − 𝑥𝑥𝑖𝑖 ) + 𝐵𝐵𝑘𝑘𝑘𝑘 (𝑦𝑦 − 𝑦𝑦𝑖𝑖 )

Lastly, rotation vector 𝑩𝑩𝜑𝜑 has six components

𝐷𝐷1 = 𝑥𝑥𝑖𝑖 (𝑦𝑦2 − 𝑦𝑦1 ) + 𝑥𝑥2 (𝑦𝑦1 − 𝑦𝑦𝑖𝑖 ) + 𝑥𝑥1 (𝑦𝑦𝑖𝑖 − 𝑦𝑦2 ) 𝐷𝐷2 = 𝑥𝑥𝑖𝑖 (𝑦𝑦3 − 𝑦𝑦2 ) + 𝑥𝑥2 (𝑦𝑦𝑖𝑖 − 𝑦𝑦3 ) + 𝑥𝑥3 (𝑦𝑦2 − 𝑦𝑦𝑖𝑖 ). 𝐷𝐷3 = 𝑥𝑥𝑖𝑖 (𝑦𝑦3 − 𝑦𝑦1 ) + 𝑥𝑥1 (𝑦𝑦𝑖𝑖 − 𝑦𝑦3 ) + 𝑥𝑥3 (𝑦𝑦1 − 𝑦𝑦𝑖𝑖 )

(9)

𝐵𝐵𝜑𝜑1 = −

where 𝑘𝑘 = 1,2,3. Relative displacements in external nodes are counted from one to six and correspond to Δ𝑢𝑢1 = δ𝑢𝑢1𝑥𝑥 , Δ𝑢𝑢2 = δ𝑢𝑢1𝑦𝑦 , Δ𝑢𝑢3 = δ𝑢𝑢2𝑥𝑥 and so on, (Fig. 3).

𝐵𝐵𝜑𝜑4

=−

sin 𝛼𝛼1

𝑟𝑟1 cos 𝛼𝛼2 𝑟𝑟2

; 𝐵𝐵𝜑𝜑2 = − ; 𝐵𝐵𝜑𝜑2

=−

cos 𝛼𝛼1

𝑟𝑟1 sin 𝛼𝛼3 𝑟𝑟3

; 𝐵𝐵𝜑𝜑3 = ; 𝐵𝐵𝜑𝜑3

=

sin 𝛼𝛼2

𝑟𝑟2 , cos 𝛼𝛼3

12)

(13)

(14)

𝑟𝑟3

where the angles and radii are shown in Fig. 3. Nodal forces in each sub-element are 𝒇𝒇𝑘𝑘𝑖𝑖𝑖𝑖 = �𝑩𝑩𝑇𝑇 ∙ 𝝈𝝈 + 𝑩𝑩𝑇𝑇𝜑𝜑 𝜇𝜇 𝛽𝛽𝜑𝜑 𝜀𝜀𝜑𝜑𝑖𝑖 � 𝑆𝑆𝑖𝑖𝑘𝑘 ,

(15)

where surface area of the k-element is 𝑆𝑆𝑖𝑖𝑘𝑘 . We collect forces from the sub-elements and obtain complete description of nodal forces in each discrete element. 4.2 Elastic solutions First, we test the method assuming that the material exhibits elastic properties only, where the equation of state is not included. Elastic constants are listed in Table 1. The analysis is developed for a plane strain problem, as shown in Fig. 4. Figure 3: Discrete element is divided into three subelements. Material properties are defined in mass centers of the sub-elements. Components of deformation matrices 𝑩𝑩 are listed below. Sub-element 1: 1 𝐵𝐵11 =−

𝑦𝑦2 −𝑦𝑦𝑖𝑖

1 𝐵𝐵31

𝐷𝐷1

=

1 ; 𝐵𝐵13 =

1 𝐵𝐵22

2

;

1 𝐵𝐵32

Sub-element 2: 2 𝐵𝐵13 =−

𝑦𝑦3 −𝑦𝑦𝑖𝑖 𝐷𝐷2

2 𝐵𝐵33 =

=

2 ; 𝐵𝐵15 =

2 𝐵𝐵24

2

𝑦𝑦1 −𝑦𝑦𝑖𝑖 1 𝐵𝐵11

2

𝑦𝑦2 −𝑦𝑦𝑖𝑖

2 ; 𝐵𝐵34 =

Sub-element 3:

𝐷𝐷1

𝐷𝐷2

2 𝐵𝐵13

2

1 ; 𝐵𝐵22 =

;

1 𝐵𝐵33

=

2 ; 𝐵𝐵24 = 2 ; 𝐵𝐵35 =

𝑥𝑥2 −𝑥𝑥𝑖𝑖 𝐷𝐷1

1 𝐵𝐵24

2

;

1 ; 𝐵𝐵24 =−

1 𝐵𝐵34

𝑥𝑥3 −𝑥𝑥𝑖𝑖 𝐷𝐷2 2 𝐵𝐵26 2

=

1 𝐵𝐵13

𝑥𝑥1 −𝑥𝑥𝑖𝑖

2

2 ; 𝐵𝐵26 =−

2 ; 𝐵𝐵36 =

2 𝐵𝐵15

2

𝐷𝐷1

𝑥𝑥2 −𝑥𝑥𝑖𝑖 𝐷𝐷2

(10)

. (11)

Figure 4: Block of concrete (10𝑐𝑐𝑐𝑐 × 40𝑐𝑐𝑐𝑐) is subjected to horizontal dynamic impulse. The plate is discretized using 1600 triangle elements and each element is further sub-divided into three elements.

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Figure 5: Block of concrete is subjected to horizontal impact (blue line). Displacement 𝑢𝑢𝑥𝑥 (red line) are measured in the central point of the sample (red dot). Displacement contours in x- and y-directions are depicted at points 1, 2 and 3. Table 1: Elastic properties 𝐵𝐵

𝐺𝐺𝐺𝐺𝐺𝐺 30

𝜇𝜇

𝛽𝛽𝜑𝜑

26

2

𝐺𝐺𝐺𝐺𝐺𝐺

−

𝜌𝜌

𝑝𝑝𝐸𝐸𝐸𝐸𝐸𝐸

𝑛𝑛𝐸𝐸𝐸𝐸𝐸𝐸

2400

1.1

2⁄3

𝑘𝑘𝑘𝑘⁄𝑚𝑚3

𝐺𝐺𝐺𝐺𝐺𝐺

of 270𝑀𝑀𝑀𝑀𝑀𝑀, we assume that the rebar responds only in an elastic manner. Strength of selected concrete in compression at strain rate 1/s is 100MPa and in tension is ten times smaller. Furthermore, we assume that fracture energy 𝐺𝐺𝑓𝑓 is stochastic. This means that the fracture energy in each element is drawn randomly from a normal distribution. Standard deviation of the distribution is 0.8 𝐺𝐺𝑓𝑓 . It is also assumed that this energy cannot be smaller than 0.2 𝐺𝐺𝑓𝑓 . The sample is discretized using 800 triangle elements.

−

The block of concrete is discretized using 1600 triangle elements, where each element is subdivided into three elements. A dynamic impulse (blue line) is applied as shown in Fig. 5. The displacement contours in x- and y-directions are presented at points 1, 2 and 3. The points are marked on the plot of displacements 𝑢𝑢𝑥𝑥 versus time (red line). These displacements are measured in the central point of the sample (red dot). The dynamic impulse creates shear perturbations along free surfaces. Consequently, as time progresses, vertical displacements produce increasingly complex patters. 4.3 Concrete block: Elastic-plastic solution with fracture In the next test, the concrete block is subjected to compression with constant velocity 10m/s, Fig. 6. In this simulation, loading continues until the sample becomes fully damaged, i.e. the sample’s load bearing capacity is reduced to zero. Note that lateral displacements are not permitted along the top and bottom surfaces. As shown, the sample is 20cm toll and 10cm wide. We introduce a stainless steel rebar (steel 304L) in the middle of the sample, where elastic properties of the rebar are 𝐵𝐵 = 151 𝐺𝐺𝐺𝐺𝐺𝐺, 𝜇𝜇 = 81 𝐺𝐺𝐺𝐺𝐺𝐺 and mass density is 𝜌𝜌 = 8030𝑘𝑘𝑘𝑘/𝑚𝑚3 . The rebar has a rectangular shape (1.4 × 1.4𝑐𝑐𝑐𝑐). Since yield stress of the steel is in the range

Figure 6: Plane strain block of concrete is subjected to constant velocity 10m/s. Top and bottom surfaces are laterally constrained. The sample is discretized using 800 triangle elements. The red area located in the middle of the sample represents stainless steel rebar. The rebar is constructed using 8 elements.

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Report ARA: 5 September 20, 2018, Alek & Research Associates, LLC and University of New Mexico

Figure 7: Average stress calculated at the top of the block is plotted as a function of time. Damage counters are presented at instances corresponding to points 1 to 5. Material responses are monitored in all subelements and, as a result, we have 2400 integration points. An average normal stress is calculated along the top surface. The average stress is plotted as a function of time (red line), Fig. 7. The damage progression is depicted at points from 1 to 5. The normalized damage (𝜂𝜂 ⁄𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 ) takes values from zero (no damage) to one, where the fracture process is complete. Initially, damages are formed in the upper corners of the sample, and then, the damages are spreading down into the sample. The rebar (red square) is placed on the plots as well. As expected, the upper part of the sample experiences heavy damages. Also, lateral deformation is significant. Lastly, we monitor element rotations. The rotations become significant in the upper corners of the samples and their magnitudes are in the range 𝜋𝜋⁄3. Once again, all parameters of the viscoplastic model are listed in the Table 2.

see damage free domains, where large chunks of concrete remain intact. We also notice that, the rebar shields the material from incoming waves.

Table 2: Viscoplasticity and fracture

Originally, the deformable discrete element method was used to study fracture processes at quasi-static conditions. In these applications, forces were equilibrated in external nodes with the use of an iterative procedure. Consequently, the method was called an iterative method of finite elements. In here, we achieve dynamic equilibrium by solving equations of motion in external and internal nodes, where mass is redistributed between these nodes. Various simulations were conducted and we concluded that our explicit time integration scheme produces consistently good results. This is true for elastic materials and the solving scheme works equally well for systems, where materials exhibit complex elastic-plastic-fracture behaviors.

𝜎𝜎𝑝𝑝0

Λ0

140

6.7 0.5 0.96 600

𝑀𝑀𝑀𝑀𝑀𝑀 −

𝑛𝑛𝑝𝑝 −

𝜔𝜔𝑝𝑝 −

𝑝𝑝𝑞𝑞0

𝑛𝑛𝑞𝑞

𝑀𝑀𝑀𝑀𝑀𝑀 −

𝑞𝑞0

𝜉𝜉𝑐𝑐0 −

𝜉𝜉𝑐𝑐1 −

0.5 0.66 0.5 0.1

𝐺𝐺𝑓𝑓

𝑛𝑛𝑓𝑓

80

2

5 CONCLUSIONS We have shown that the deformable discrete element method combines properties of finite element method and particle method. The rigid motion of particles is defined in mass centers. Particle deformation is monitored in terms of relative displacements (displacements in external nodes minus the contribution of rigid translations and rotation). The method is designed for pseudoCosserat media and, in the simplified version, properly describes classic continuum. Our observations suggest that rotations are most pronounced near free surfaces, interfaces and are pronounced in damage zones.

𝐾𝐾𝐾𝐾⁄𝑚𝑚3 −

The simulations are repeated three times, Fig. 8. The stress plots exhibit similar trends. However, the mechanisms of final fracture are not quite the same. In all the cases, cones of undamaged material are present near the top and bottom surfaces. The damages are localized within certain zones. Also, we

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Report ARA: 5 September 20, 2018, Alek & Research Associates, LLC and University of New Mexico

Figure 8: Plots of stress versus time are constructed for three random realizations of fracture energy 𝐺𝐺𝑓𝑓 . Damage maps are shown at the final stage of the damage process (point P). The damage is measured from zero to one. Also, the damage is directional. Mech (ASCE) 1987; 113(11): DOI: 10.1061/(ASCE)0733-9399(1987)113:11(1619).

We emphasize that the method is easily extendable to account for the crack growth along particle interfaces. Also, particle can be easily separated from each other.

5. Zubelewicz A, Mroz Z. Numerical simulation of rock burst process treated as problems of dynamic instability. Rock Mech Rock Eng 1983 (4): 253-274. DOI: 10.1007/BF01042360.

Over so many years, increasingly sophisticated features have been adapted to finite element codes. Finite elements can be separated from each other, particles can be created, and cracks are allowed to propagate along interfaces. At the same time, element deletion helps in solving issues of excessive deformation and fracture. It is interesting that the long forgotten DDEM method combines the FEM and DEM features in a mutually consistent manner.

6. Zubelewicz A, O’Connor K , Dowding CH, et al. A constitutive model for the cyclic behavior of dilatant rock joints. 2nd International Conference on Constitutive Laws for Engineering Materials: Theory and Applications, Eds. C.S. Desai, E. Krempl, P.D. Kiousis and T. Kundu, January 5-8; 1987, Tuscon, Arizona.

REFERENCES

7. Zubelewicz A., A certain version of finite element method, Ph.D. dissertation,Warsaw Technical University; 1979, Warsaw, Poland.

1. A. Zubelewicz A. Iterative method of finite elements. 2nd Conf. on Computational Methods in Mechanics of Structures 1975; Gdansk, Poland.

8. Kawai T, Kondou K. New beam and plate bending elements in finite element analysis. 1976, J Seisan Kenkyu, Institute of Industrial Science, University of Tokyo; 28: 409 – 412.

2. Zubelewicz A. Iterative method of finite elements in analysis of brittle materials. 3rd Conf. on Computational Methods in Mechanics of Structures 1977; Opole, Poland.

9. Kruszewski J, Gawroński W, Wittbrodt E, et al. Metoda sztywnych elementów skończonych, (Rigid finite element method). Arkady 1975.

3. Zubelewicz A. A proposal of new structural model of concrete. Arch Civil Eng 1983:416-439. 4. Zubelewicz A, Bazant, ZP. Interface element modeling of fracture in aggregate composites. J Eng

10. Kawai T. New discrete models and their application to seismic response analysis of structures, Nuclear Eng Design 1978; 48: 207-229.

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Report ARA: 5 September 20, 2018, Alek & Research Associates, LLC and University of New Mexico

11. Cundall PA. (1971) A Computer Model for Simulating Progressive Large Scale Movements in Blocky Rock Systems. Proc Symposium of the International Society for Rock Mechanics, Society for Rock Mechanics (ISRM) 1971 France, II-8. 12. Cundall PA (1979). A discrete numerical model for granular assemblies. Géotechnique 1979; 29 (1): 47–65. 13. Rojek J, Zubelewicz A, Madan N, et al., New formulation of the discrete element method. AIP Conference; 2018:1922, 030009 (2018), doi.org/10.1063/1.5019043. 14. Zubelewicz A, Precursors of dynamic excitations and rupture in rocks. 2016; 50(6): DOI: 10.1007/s00603-017-1167-5.

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