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Computer Methods in Applied Mechanics and Engineering manuscript No.

(will be inserted by the editor)

DP-MPM: Domain partitioning material point method for evolving

multi-body thermal-mechanical contacts during dynamic fracture and

fragmentation

Mian Xiao ·Chuanqi Liu ·WaiChing Sun

the date of receipt and acceptance should be inserted later

Abstract We propose a material point method (MPM) to model the evolving multi-body contacts due to

crack growth and fragmentation of thermo-elastic bodies. By representing particle interface with an im-

plicit function, we adopt the gradient partition techniques introduced by Homel and Herbold (2017) to

identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate

the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the

fragmentation process. By storing the temperature at material points, the resultant MPM model captures

the thermal advection-diffusion in a Lagrangian frame during the fragmentation, which in return affects

the structural heating and dissipation across the frictional interfaces. The resultant model is capable of

replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures

or insertion of interface elements. A staggered algorithm is adopted to integrate the displacement and

temperature sequentially. Numerical experiments are employed to validate the diffusion between the ther-

mal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes

affect the path-dependent behaviors of the multi-body systems.

Keywords material point method; multi-body contact; fragmentation; brittle fracture; thermomechanics

1 Introduction

Path-dependent behaviors of particulate systems manifest from both the micro-mechanical responses

within individual particles (Zhang et al.,1990;Cil and Alshibli,2012;Na et al.,2017;Liu and Sun,2020c)

and how these particles interact with the surrounding particles over time (Marone and Scholz,1989;Sun

et al.,2013;Kuhn et al.,2015;Liu et al.,2016;Wang and Sun,2017a;Gupta et al.,2019;Bryant and Sun,2019;

Wang and Sun,2019;Wang et al.,2021). While simulations conducted via discrete/distinct element meth-

ods may update particle contact connectivity at each incremental time step, the contact laws (e.g. Herizan

contact (Johnson,1982), linear frictional model (Cundall and Strack,1979)) employed in a discrete element

method are highly idealized. Examples of these idealizations include assuming speciﬁc particle shapes

(e.g. sphere, ﬂat half-space), the geometry of contacts (e.g. point-wise contacts, overlapped domain), and

topology (e.g. two-particle contact, neglected deformation of particles). In recent decades, many studies are

dedicated to analyzing how these idealizations affect the accuracy of the predictions and propose reme-

dies to overcome the limitations. For instance, contacts of non-spherical particles are often approximated

by clustering spheres together to form regular shapes (ˇ

Smilauer and Chareyre,2010). Alternatively, Boon

et al. (2012) and Kawamoto et al. (2016) introduce implicit functions to represent the geometry of particles

of different shapes.

Corresponding author: Chuanqi Liu

Former postdoctoral research scientist, Department of Civil Engineering and Engineering Mechanics, Columbia University,

614 SW Mudd, New York, NY, 10027, now Associate Professor, the State Key Laboratory of Nonlinear Mechanics, Institute of

Mechanics, Chinese Academy of Sciences Beijing, China, 100090. E-mail: chuanqil@imech.ac.cn

2 Mian Xiao et al.

Nevertheless, one key issue for these attempts is that the stress ﬁeld inside individual particles is not

available. Hence, alternative theories that are based on force or homogenized stress of individual par-

ticles proposed to idealize the damage process either as an instant splitting of individual particles (e.g.

Harmon et al. (2020)), an instant debonding of prepackaged agglomerates of particles (e.g. Cheng et al.

(2004)), or removal of particles (e.g. (Wang et al.,2008)). While these methods may replicate some geomet-

rical and topological features due to grain crushing and fragmentation, the instant splitting or debonding

idealization is not suitable for high-strain rate impact where the crack growth and propagation speed are

important. Furthermore, since crack and damage is not triggered not simply by the magnitude of the ho-

mogenized stress of a particle but also how strain energy stored and concentrated locally, the lack of stress

and strain ﬁeld of the grain in classical discrete element simulations may lead to a violation of fracture

mechanics principles even in the brittle regimes.

Furthermore, the lack of rate-dependence in the breakage criteria also make those predictions not suit-

able for dynamic simulations where mechanics of fragmentation and damage of brittle grains can be trig-

gered by fracture and crack branching and hence highly sensitive to loading rates (Yoffe,1951;Congleton

and Fetch,1967), the microstructures and the spatial heterogeneity of individual particles (Ravi-Chandar

and Knauss,1984). To circumvent these limitations, previous works such as Liu and Sun (2020a) explored

the possibility of simulating granular assembles as a multi-body contact in an implicit quasi-static regime.

While path-dependent behaviors could be triggered by rearrangement of particles and the dissipation due

to the frictional slip, the topological changes of the particles due to fracture, damage, and fragmentation

are not considered.

The purpose of this paper is to ﬁll this knowledge gap by proposing a material point framework suit-

able to model the multi-body contact thermo-mechanics of assemblies composed of brittle particles. To

capture the rate-dependence and temperature-dependence of the frictional contact and fracture of the par-

ticles, we present a material point method that captures the thermo-mechanical coupling for both bulk

materials and contact surfaces, incorporate the domain partitioning techniques originated from Homel

and Herbold (2017) to handle the continuously evolving contact geometry, and introduce constitutive laws

to capture the thermal-mechanical frictional contact, damage laws, interfacial thermal conduction among

particles in the ﬁnite deformation regime. Consequentially, the proposed model is a capture of replicating

the dynamic fracture and fragmentation while handing the evolving contacts due to evolving geometrical

and topological changes of the crushed particle assembles.

The rest of this paper is organized as follows. We will ﬁrst provide a brief literature review on topics

relevant to the multi-body thermal-mechanical contact problem in this Section. Section 2then lists the pre-

sumptions and derives a set of governing equations starting from a free energy representation. Section 3

provides details on the implementation of the proposed model in the MPM. Section 4presents numeri-

cal examples for the validation and demonstration of the capacity of the model. Concluding remarks are

provided in Section 5.

1.1 Literature review on dynamic fracture simulations

To simulate the evolving contacts properly, one must replicate the deformation, fracture, and dam-

age that trigger the topological and geometric changes of the interfaces. For high-strain-rate applications

where the crack may propagate faster than the Rayleigh wave, the crack patterns are highly sensitive to

the loading rate as crack branching occurs that changes how energy dissipates (Ravi-Chandar and Knauss,

1984). Furthermore, the loading rate also plays an important role in the thermal-mechanical responses of

a path-dependent material. While damage, plasticity and elastic structural heating may all lead to heat,

the local temperature changes and the subsequent heat transfer may also trigger different readjustment

of deformation pathways and affect the macroscopic mechanical responses (Rittel et al.,2017;Zubelewicz,

2019;Lieou and Bronkhorst,2021).

At the high-strain-rate regime, the local temperature may change near the crack tips or contact areas

due to elastic/inelastic structural heating and dissipation that occurs almost adiabatically. This thermal

effect could be signiﬁcant enough to affect the mechanical responses, including the onset and propagation

of crack(s). For instance, Nowinski (1978) stated that a solid body subject to elastic deformation may heat

up in a compressive region and cool down in a tensile region. It is also experimentally observed in rocks

MPM Thermo-Contact 3

that temperature rises in the regions with intensive stress concentration and drops for stress relaxation

(Liu et al.,2004,2007). Meanwhile, temperature also plays a profound role in the frictional coefﬁcient of

the contacts and affects the strength and ductility (Paterson and Wong,2005).

An important prerequisite to capture these thermo-mechanical coupling effects is the precise repre-

sentation of the interface geometry. This can be achieved by either embedding strong discontinuity in the

interpolated displacement ﬁeld or via a smeared crack approach where a continuous indicator function is

used to approximate the sharp discontinuity. The extended ﬁnite element method (X-FEM) and the gen-

eralized ﬁnite element method (GFEM) belong to the ﬁrst strategy. These two methods are synonymous

(Belytschko et al.,2009) and both rely on the partition-of-unity enrichment to enrich the displacement ﬁeld

(Melenk and Babuˇ

ska,1996). The crack thus can propagate independently on the mesh, which overcomes

the constraint of the cohesive models (Hillerborg et al.,1976;Xu and Needleman,1994). Nevertheless, for

dynamic cases where crack branching may occur, the generation of the enrichment function to replicate

the geometry and the integration of the residual are both non-trivial (Armero and Linder,2009;Linder and

Raina,2013). Even these technical issues can be resolved, modeling the fragmentation via the embedding

strong discontinuity remains difﬁcult due to the lack of established predictive theory for the onset and the

mode of the crack branching (Linder and Armero,2009).

Smeared crack approaches such as phase ﬁeld fracture (e.g. Miehe et al. (2010); Borden et al. (2012);

Choo and Sun (2018); Bryant and Sun (2018); Na and Sun (2018); Bryant and Sun (2021)), nonlocal or

gradient damage models Geers et al. (1998); Baˇ

zant and Jir´

asek (2002); Liu and Sun (2020c) provide an

alternative to capture the crack branching process without requiring additional criterion to predict the

onset of crack branching and the additional implementation effort to embed discontinuity. This ease of

implementation provides a great advantage in handling the evolving interfaces.

To handle the geometrical nonlinearity during fracture, Moutsanidis et al. (2019) incorporate the phase

ﬁeld fracture model in a material point method (MPM). Meanwhile, Zhang et al. (2020) enhance the MPM

with eigenerosion (Pandolﬁ and Ortiz,2012;Li et al.,2015;Wang and Sun,2017b;Qinami et al.,2019) to

simulate dynamic fracturing. Recently, Homel and Herbold (2017) employ a damage scalar ﬁeld to present

the fractures in the material point method and use this scalar ﬁeld to detect contacts. Nevertheless, the

thermal-mechanical coupling effect on path-dependent behaviors such as crack growth, frictional slip, and

thermal conduction across the interface has not yet been considered.

1.2 Literature review on computational contact thermo-mechanics

If a mode II, mode III or mixed-mode crack is propagating under compression or an existing crack is

closed, then the frictional contact may introduce sufﬁcient energy dissipation that alters the fracture pat-

terns. For high-strain-rate applications, the dissipation due to friction may lead to a substantial amount

of heat building up near the adiabatic limit and hence affect the path-dependent behaviors of the solids

Bryant and Sun (2021); Sun (2015); Khoei and Bahmani (2018). Meanwhile, at the low-strain-rate regime,

the thermal conductance across contact boundaries must be replicated in order to capture the interplay

between the fracture process and thermo-mechanical contact mechanics. In the fully coupled thermome-

chanical setting, the heat conductance across the contact boundary depends on the normal contact pres-

sure, while the contact friction may also decrease as a result of temperature rise (Johansson and Klarbring,

1993). Hence, capturing both the contact conductance and normal contact pressure is of critical impor-

tance for precisely replicating the thermo-mechanical contacts. For simplicity, the linear constitutive law

for pressure-dependent thermal conductance has been widely used in the literature (e.g. Johansson and

Klarbring (1993); H¨

ueber and Wohlmuth (2009)). Meanwhile, Wriggers and Miehe (1994) propose a more

sophisticated power law to describe contact conductance. In these cases, ﬁnite element solvers are often

the choice to simulate the contact thermo-mechanical problems.

Early works on the thermo-elastic contact with FEM usually implemented a node-to-segment contact

algorithm, where the contact constraints are enforced via a penalty formulation (Johansson and Klarbring,

1993;Wriggers and Miehe,1994;Pantuso et al.,2000). In the last decade, the mortar method provides

a variationally consistent way for contact discretization (Temizer,2014;Dittmann et al.,2014). Contact

algorithms with the mortar method can enforce the contact constraint exactly via a Lagrange multiplier

and therefore improves accuracy as well. In addition to modeling the thermo-elastic contact in the elastic

4 Mian Xiao et al.

regime, more recent work, such as Seitz et al. (2018), has also formulated Nitsche’s method to simulate

thermal-elasto-plasticity contact problems in a ﬁnite element solver. Nevertheless, to the best knowledge

of the authors, there has not yet been any attempt to resolve the thermo-mechanical contact problems with

evolving interfaces due to the brittle fracture via MPM.

2 Thermo-mechanical contact mechanics with evolving contacts

This section presents the theory of the thermo-mechanical contact mechanics for simulating the path-

dependent responses of multiple continuum bodies in the geometrically nonlinear regime. Sections 2.1 and

2.2 describe the constitutive laws that capture the interplay among the frictional heating, surface, and bulk

conduction and the evolution of brittle damages and the evolving interfaces. The balance principles that

serve as the constraints for the material point model are provided in Section 2.3. In addition, the following

assumptions are made to simplify the formulation.

1. We assume the effective stress theory (cf. Simo and Ju (1987); Lemaitre (1985)) is valid such that the

stored elastic energy of a material point representing a representative elementary volume (REV) is

related to that of a ﬁctitious pure elastic body by a degradation function.

2. The thermo-mechanical coupling effects are addressed by introducing a coupling function inﬂuenced

by both mechanical ﬁelds and thermal ﬁelds. In this sense, the free energy function consists of three

parts: (1) the pure mechanical storage energy; (2) the pure thermal energy; (3) the thermo-mechanical

coupling energy (Simo and Miehe,1992).

3. We assume the mechanical damage inﬂuences the mechanical and thermo-mechanical coupling free

energy contributions but does not affect the heat capacity.

4. To calibrate the energy consumed during the fracture propagation, we assume all of the energy released

in the damage evolution process is equal to the crack surface energy (Oliver,1995), which is reasonable

for brittle fracture.

2.1 Thermoelastic constitutive framework for the bulk continuum

Here we deﬁne the constitutive law for bodies in contacts. The continuum bodies are thermally con-

ductive and may exhibit damage and degradation. Based on the effective stress theory, we assume that

the constitutive responses of the materials in the bodies can be characterized by applying a degradation

function to the hyperelastic energy functional of a ﬁctitious material that exhibits no damage. As such, the

corresponding free energy function per unit mass ψcan be decomposed into three components, i.e.,

ψ(b,d,θ) = ψe(b,d) + M(J,d,θ) + ψt(θ), (1)

where b=FF Tis the left Cauchy-green strain tensor, dis the scalar damage, θis the temperature, and

J=det Fis the determinant of F. As such, ψe(b,d)is the mechanical contribution, M(J,d,θ)is the thermo-

mechanical coupling term, and ψt(θ)is the thermal contribution (see also Wriggers and Miehe (1992);

Wriggers and Zavarise (2004)). Following Simo and Miehe (1992), the expressions for these terms are:

ρoψe(b,d) = (1−d)(U(J) + W(biso )), (2)

ρoM(J,d,θ) = −3(1−d)αt(θ−θ0)∂U

∂J, (3)

ψt(θ) = c[(θ−θ0)−θlog(θ/θ0)], (4)

respectively, where ρois the mass density on the unit reference volume, biso =J−2/3brepresents the iso-

choric part of the left Cauchy-Green strain tensor b,U(J)is the volumetric part of the elastic stored energy,

W(biso)is the isochoric counterpart of the undamaged material, αtis the thermal expansion coefﬁcient, cis

the heat capacity per mass, and θ0is the reference temperature. The energy contributions of the damaged

MPM Thermo-Contact 5

real material (ψeand M) and those of the undamaged ﬁctitious material (ψeand M)are related by the

degradation term (1−d), i.e.,

ψe(b,d) = (1−d)ψe(b),M(J,d,θ) = (1−d)M(J,θ), (5)

where ρoψe=U(J) + W(b)indicates the undamaged storage energy on the unit reference volume only de-

pending on deformation (Murakami,2012), ρoM(J,θ) = −3αt(θ−θ0)∂U

∂Jindicates the undamaged thermo-

coupling energy component on the unit reference volume.

The Kirchhoff stress τ, the entropy ηand the storage energy release Yon the unit reference volume

corresponding to the free energy reads (Marigo et al.,2016),

τ=2ρo∂ψ

∂bb,η=−∂ψ

∂θ ,Y=−ρo∂ψ

∂d. (6)

We also extract the damage multiplier as what (5) does:

τ= (1−d)τ,τ:=2ρo ∂ψe

∂b+∂M

∂b!b, (7)

where τis called the effective Kirchhoff stress in continuum damage mechanics (Chaboche,1988). This

extraction is helpful since τindicates the Kirchhoff stress without mechanical damage softening.

2.2 Thermal-sensitive frictional constitutive framework for contacts

Consider a contact between two distinct bodies. The Karush-Kuhn-Tucker (KKT) conditions for the

normal and tangential contact on the contact boundary Γcreads (H¨

ueber and Wohlmuth,2009;Liu and

Sun,2020a):

pcn <0, δn>0, pcn δn=0. (8)

|pct| ≤ µc|pcn |,(˙

δt=0 for |pct|<µc|pcn |,

|˙

δt| ≥ 0 for |pct |=µc|pcn |.(9)

where pcn :=pc·n,pct :=pc·tindicates the normal and tangential contact stresses with pcthe traction on

Γcand n,tthe normal vector and right-handed tangential vector for the contact surface; δnand δtdenote the

magnitude of the inter-surface gap in the normal and tangential direction respectively; µcis the frictional

coefﬁcient; the dots in (9) denote the time derivative, hence ˙

δtindicates the rate of relative slip between

two body ﬁelds along the direction of t.Speciﬁcally, we refer to δtas:

˙

δtk =(vt2−vt1, for k=1,

vt1−vt2, for k=2. (10)

where the subscript kis an index for the pair of bodies in contacts (cf. H¨

ueber and Wohlmuth (2009)); vtk is

the velocity component at the location of contact along the tangential direction tfor different bodies.

For non-slip contact problems, the contact boundary can be viewed as a conductive boundary in terms

of thermal effects, where the normal heat ﬂux is expressed as a multiple of the surface conductance hcand

the temperature jump [θ]between two surfaces (Wriggers and Miehe,1994):

qck =hc[θ]k,[θ]k=(θ2−θ1, for k=1,

θ1−θ2, for k=2. (11)

where qcindicates the heat ﬂux across Γc. The constitutive law for the contact surface conductance hcreads

(Wriggers and Zavarise,2004),

hc=h0

pcn

p0

e

(12)

6 Mian Xiao et al.

where p0>0 is called vicker hardness as a regularization of the normal contact stress, h0is a reference

conductance at |pcn |=p0,eis a material parameter indicating the power scale with respect to pcn. Notice

that by picking e=1, this law is further simpliﬁed into a linear relationship

hc=¯

γ|pcn|,¯

γ:=h0

p0

. (13)

where ¯

γis the stress-conductance coefﬁcient deﬁned as the ratio between reference conductance and the

vicker hardness, which can be regarded as a material property.

In frictional contact cases, we need to take into account the heat generated during slipping. We assume

that the surface speciﬁc heat is zero and heat dissipates into both ﬁelds equally. Then the normal heat ﬂux

in (11) is modiﬁed as (Rieger and Wriggers,2004):

qck =hc[θ]k+1

2pct ˙

δt(14)

We also assume that the frictional coefﬁcient on the contact surface is temperature-dependent, which

has the following form (Johansson and Klarbring,1993):

µc=µco

<θdam −max(θ1,θ2)>2

(θdam −θref)2(15)

where θdam indicates the temperature where the frictional response on the contact surface completely dis-

appears; θref is a reference temperature constant, usually chosen as the initial temperature; µco is the un-

damaged frictional coefﬁcient at the reference temperature. <·>are the Macaulay brackets such that

<x>= (x+|x|)/2

2.2.1 Speciﬁc constitutive relationships

We adopt the compressible Neo-Hookean hyperelastic model to replicate the elastic responses of the

brittle materials. This hyperelasticity energy consists of two components that calculate the elastic stored

energy due to volumetric and deviatoric deformation, i.e., (Ogden,1997):

U(J) = 1

2K(log J)2, (16)

W(biso) = 1

2G(tr(biso)−3). (17)

where Kand Gare the bulk modulus and the shear modulus respectively. tr(·)is the tensor trace opera-

tion. For the thermal ﬁelds, we consider the isotropic Fourier’s heat conduction law in the Eulerian form

(Nowinski,1978):

q=−κ∇xθ, (18)

where κis the heat conduction coefﬁcient. ∇xis the gradient operator on the deformed conﬁguration.

To capture the fracture propagation with damage evolution, we adopt Rankine’s damage model with

linear strain-softening for brittle damage (Rankine’s rotating crack model) (Rockﬁeld,2007). In this theory,

the damage evolution is governed by the largest effective principal tensile stress σmassuming that the

damage yield function takes the form Φ(σm,R) = σm−R(Cervera et al.,1995), where Ris the damage

internal variable. Φand Rshould satisfy the KKT condition, which is addressed as:

Φ≤0, ˙

R≥0, Φ˙

R=0. (19)

We assume that the damage is initially zero and starts evolving once σmreaches the critical stress

threshold σf. This makes it possible to derive an explicit functional representation of the damage internal

variable: R=max(σf, maxt(σm)), where maxt(·)indicates ﬁnding the largest historical value for a speciﬁc

term. With this knowledge, Rcan be interpreted as the largest effective principal tensile stress in the stress

history of a particular material point.

MPM Thermo-Contact 7

To complete the formulation of this damage constitutive model, the mapping function from the internal

variable Rto the damage scale dmust be deﬁned. This is derived from the linear strain-softening after the

crack evolution is triggered (Oliver,1995):

d(R) = ((1+Hs)(1−σf

R),σf<R<σf(1+1

Hs),

1 , R≥σf(1+1

Hs).(20)

where Hs:=Hslch/(1−Hslc h)is the mesh-regularized damage modulus with lch indicating the mesh

characteristic length (Cervera and Chiumenti,2006) and Hsstanding for the brittleness factor. lch is often

chosen as the diagonal length of a cell in a structured grid, and Hscan be derived from the material

properties with Assumption 4 in Section 2.1 (for more details, see Cervera and Chiumenti (2006)):

Hs=σ2

f

2EG f

(21)

where Eis Young’s modulus and Gfis the critical energy release rate.

2.3 Governing equations for the thermo-mechanical problems

Here we brieﬂy review the balance of mass, linear momentum, and energy for the contact problems.

The balance of mass reads:

ρ=ρo/J, (22)

where ρand ρoare the mass densities on the deformed and reference volume accordingly. In the current

conﬁguration, the balance of linear momentum reads:

ρa=∇x·σ+ρg, (23)

where ais the acceleration, gis the body force per mass, σ=1

Jτrepresents the Cauchy stress, ∇x·indicates

the divergence on the deformed conﬁguration. An important advantage of the MPM formulation is that

the grid used to compute the residual is also doubled as the updated Lagrangian frame and hence the

convection term in the acceleration in the Eulerian frame does not appear (Sulsky et al.,1994).

To formulate the balance of energy, we ﬁrst deﬁne the internal energy per unit mass as e:=ψ+θη.

The rate of internal energy change consists of external stress power and the incoming heat ﬂux (we ignore

external heat sources). Hence, the local form of the energy balance equation reads,

ρ˙

e=σ:d− ∇x·q(24)

where d:=1

2(l+lT)is the symmetric part of the velocity gradient l=∇xvwith vrepresenting velocity

and qindicating the heat ﬂux on the unit deformed volume. Recall that the rate of the change of internal

energy is related to those of the free Helmholtz energy, temperature, and entropy, i.e.,

˙

e=˙

ψ+˙

θη +θ˙

η, (25)

where the time derivative of ψ(b,θ,d)is

˙

ψ=∂ψ

∂b:˙

b+∂ψ

∂θ ˙

θ+∂ψ

∂d˙

d, (26)

Substituting η=−∂ψ/∂θ into (26), we have

˙

ψ+˙

θη =∂ψ

∂b:˙

b+∂ψ

∂d˙

d, (27)

Since we only have the damage scalar as the internal variable in this formulation, we can deﬁne the

internal energy dissipation following Holzapfel (2000):

Dint =−∂ψ

∂db,θ

˙

d=1

ρoψe−3αt(θ−θ0)∂U

∂J˙

d, (28)

8 Mian Xiao et al.

where Dint refers to the internal dissipation term for the damage internal variable d.

We further implement the following equation due to η=η(b,θ,d):

θ˙

η=θ∂η

∂b:˙

b+θ∂η

∂θ ˙

θ+θ∂η

∂d˙

d, (29)

Following Holzapfel (2000), (29) is rewritten as:

θ˙

η=cF˙

θ+He+Hin, (30)

where

cF=θ∂η

∂θ b,d

,He=θ∂η

∂bθ,d

:˙

b,Hin =θ∂η

∂db,θ

˙

d, (31)

where cFdenotes the speciﬁc heat per unit mass, Heindicates the structural heating term, induced by the

elastic volumetric deformation, and Hin is the structural heating term due to the damage internal variable.

To handle the partial derivatives w.r.t. ηin (31), we substitute the deﬁnition of ηin (6) with (1):

η=−∂ψ

∂θ =−∂M

∂θ −∂ψt

∂θ , (32)

Insert (32) to (31), and we have

cF=−θ∂2M

∂θ2−θ∂2ψt

∂θ2, (33)

He=−θ∂2M

∂θ∂b+∂2ψt

∂θ∂b:˙

b, (34)

Hin =−θ∂2M

∂θ∂d+∂2ψt

∂θ∂d˙

d, (35)

From the free energy form in Section 2.1, we observe that the coupled energy Mis linear in θ, while the

pure thermal energy ψtis independent of band d. As such,

∂2M

∂θ2=0, ∂2ψt

∂θ∂b=0,∂2ψt

∂θ∂d=0, (36)

In addition, as Mis not an explicit function of bbut an explicit function of J, we can further simplify

the partial derivative of Mwith respect to bin (34) as:

∂2M

∂θ∂b:˙

b=∂2M

∂θ∂J˙

J, (37)

Using (36) to eliminate zero terms in (33) ˜ (35) and insert (37) to (34), we derive the following expres-

sions for the practical calculation of the speciﬁc heat and the structural heating terms:

cF=−θ∂2ψt

∂θ2=c, (38)

He=−θ∂2M

∂θ∂J˙

J=1

ρo3αtθ(1−d)∂2U

∂J2˙

J, (39)

Hin =−θ∂2M

∂θ∂d˙

d=−1

ρo3αtθ∂U

∂J˙

d, (40)

We ﬁnally insert all terms in (27) and (31) into (24):

ρ∂ψ

∂b:˙

b−ρDint +ρcF˙

θ+ρHe+ρHin =σ:d− ∇x·q(41)

MPM Thermo-Contact 9

where σ:d=2ρ∂ψ

∂bb:d=ρ∂ψ

∂b:˙

b(cf. Holzapfel (2000)). As a result:

ρcF˙

θ+ρHe+ρHin =− ∇x·q+ρDint. (42)

By rearranging Eq (42), we recover the balance of energy equation in Eq. 103 of Miehe et al. (2015)

ρcF˙

θ=− ∇x·q+ρDint −ρHe−ρHi n, (43)

To complete the formulation of a thermo-mechanical boundary value problem in a deformed body Ω,

we present the boundary conditions on its boundary Γ. We decompose the mechanical boundary into three

parts as Γ=Γu∪Γt∪Γcwith Γu∩Γt=Γu∩Γc=Γt∩Γc=∅, where Γuis the displacement boundary,

Γtis the traction boundary and Γcis the contact boundary. We decompose the thermal boundary into four

parts as Γ=Γθ∪Γq∪Γh∪Γcwith Γθ∩Γq=Γθ∩Γh=Γq∩Γh=∅, where Γθis the temperature boundary,

Γqis the heat ﬂux boundary, Γhis the conductive boundary and none of these three boundaries overlaps

with Γc. Accordingly, the boundary conditions are:

(u=uon Γu,

σ·n=ton Γt,

θ=θon Γθ,

q·n=−qon Γq,

q·n=h(θ∞−θ)on Γh.

(44)

where uis the prescribed displacement on Γu;tis the applied traction on Γt;θis the prescribed temperature

on Γθ;qis the applied normal heat ﬂux on Γq;his the surface conductive coefﬁcient and θ∞is the ambient

temperature on Γh. On Γc, we enforce the mechanical contact governing equations (8) and (9) and the

contact heat conduction equation (14) instead of prescribed boundary conditions.

3 Numerical implementation with MPM

In this section, we describe the numerical implementation of the mathematical framework that captures

the interplay between evolving contacts and dynamic fracture under non-isothermal conditions. We ﬁrst

brieﬂy review the algorithm of the MPM that updates the updated Lagrangian frame with material points

and the implication of this treatment to handle the convection of heat in the ﬁnite deformation range. Then,

we elaborate on the algorithms that detect contact and calculate separation displacement among brittle

bodies for dynamic thermo-mechanics problems. We then discuss the contact calculation with a focus on

the thermal effect. The implementation algorithm is concluded at the end.

3.1 An overview of MPM

In this MPM formulation, all physical variables (such as b,θ, and d) are stored at particles as the La-

grangian description. These particles are connected with a grid using weighting functions whose augments

are the relative positions between particles and the grid. We then calculate the unbalanced variables (such

as acceleration for the balance of momentum) of nodes and project the information back to particles. After

one MPM time step, the deformed grid is reset to the initial conﬁguration as structured since it only works

to connect particles. Therefore, the framework of MPM belongs to the updated Lagrangian form with-

out any special treatment. For the sake of clarity, xrefers to the current location; the subscript iindicates

variables attribute at the discrete grid node i, and the subscript pis used to present variables at particles.

The weak form of the balance of momentum in the current conﬁguration is :

ZΩw·ρadΩ=−ZΩ∇xw:σdΩ+ZΩw·ρbdΩ+ZΓt

w·¯

tdΓ+ZΓc

w·pcdΓ, (45)

where wis the admissible test function. The discrete form of the momentum equation is (Tao et al.,2018;

Liu et al.,2018):

miai=fext

i−fint

i+fcont

i, (46)

10 Mian Xiao et al.

where mi,ai,fext

i,fint

i, and fcont

irepresent the mass, acceleration, external force, internal force and the

contact force at grid node i, respectively. These nodal attributes are calculated with the following equations:

mi=∑

p

mpSip , (47)

fext

i=∑

p

mpSip gp+ZΓt

SitdΓ, (48)

fint

i=∑

p

Vpσp· ∇xSip , (49)

fcont

i=ZΓc

SipcdΓ. (50)

where mp,Vp,gp,σprepresents the mass, deformed volume, body force, and the Cauchy stress at particle

p, respectively, the operator ∑pindicates summing the inﬂuence of the neighbor particles at a node (Homel

and Herbold,2017), Siindicates the standard grid node basis function, Sip and ∇xSi p are the weighting

function and its gradient for the mapping between node iand particle p.

Noting that Sip links the particles and the grid, we here adopt the Convected Particle Domain Inter-

polation (CPDI) to compute Sip for accuracy, as proposed in Sadeghirad et al. (2011). Assuming that the

particle domain is a parallelogram and that the deformation gradient is approximately constant over the

particle domain, the particle domain deforms according to

(r1=F·r0

1,

r2=F·r0

2,(51)

where (r0

1,r0

2)and (r1,r2)are the vectors deﬁning a particle domain at the initial and updated conﬁgura-

tion, respectively. For the standard FE 4-node (Q4) element, which is adopted in this work, the weighting

function and its gradient are

Sip ∼

=1

4hSi(xp

1) + Si(xp

2) + Si(xp

3) + Si(xp

4)i, (52)

∇xSip ∼

=1

2Vp(Si(xp

1)−Si(xp

3)rn

1y−rn

2y

rn

2x−rn

1x+ (Si(xp

2)−Si(xp

4)rn

1y+rn

2y

−rn

1x−rn

2x, (53)

where xp

αα=1, 2, 3, 4 are the positions of the corners of the parallelogram, (r1x,r1y)and (r2x,r2y)are the

components of vectors of rn

1and rn

2, respectively, and the superscript ndenotes the nth time step.

We now consider the discrete energy equation. Staring from the variation of the energy equation in

terms of temperature with an arbitrary scalar testing function w:

ZΩwρc˙

θdΩ=ZΩ∇xw·qdΩ−ZΩwHedΩ+ZΩw(Dint − Hin)dΩ+

ZΓq

w¯

qdΓ+ZΓh

wh(θ∞−θ)dΓ+ZΓc

wqcn dΓ,

(54)

Following similar logics in the derivation of (46), the form of discrete energy equation is:

Ci˙

θi=Qext

i+Qint

i+Qecpl

i+Qdcpl

i+Qcont

i, (55)

where Ciand ˙

θiindicate the heat capacity and temperature evolution rate for at grid node i;Qext

i,Qint

i,

Qecpl

i,Qdcpl

i,Qcont

irepresent the external, internal, elastic-coupling, damage-coupling, and contact thermal

MPM Thermo-Contact 11

loads at grid node i, respectively. These nodal attributes are calculated with the following equations:

Ci=∑

p

mpcpSip , (56)

Qext

i=ZΓq

Si¯

q dΓ+ZΓq

Sih(θ∞−θ)dΓ, (57)

Qint

i=∑

p

Vpqp· ∇xSip , (58)

Qecpl

i=−∑

p

VpSip(He)p, (59)

Qdcpl

i=∑

p

VpSip (Dint )p−(Hin )p, (60)

Qcont

i=ZΓc

Siqcn dΓ. (61)

where cp,θp,qpindicate the speciﬁc heat, the temperature, and the heat ﬂux at each particle p, respectively;

(Dint )p,(He)p, and (Hin )pare the internal dissipation, the elastic structural heating and internal variable

induced heating terms at each particle p.

At the end of one MPM time step, we will perform time integration on all nodal unbalanced variables

and update the material ﬁelds at particles accordingly. There are two popular time integration strategies to

solve a fully coupled system, i.e. the monolithic integration scheme and the staggered integration scheme.

The monolithic time integration requires the equations for different ﬁelds to be solved simultaneously

(Romero,2010). For this approach, an iterative strategy is usually required to determine the amount of

increment in both thermal and mechanical ﬁelds since the equations are strongly coupled together, which

increases its computational cost signiﬁcantly. In the staggered time integration, however, the entire system

is split into individual ﬁelds that perform time marching separately (Felippa and Park,1980). With the

explicit Euler time integration scheme, this approach evolves individual ﬁelds sequentially based on the

latest information from the thermal and mechanical ﬁelds (Farhat and Lesoinne,2000). As a result, the

staggered scheme is much more efﬁcient than the monolithic scheme. In this paper, we adopt the staggered

time-stepping scheme for the time integration in MPM.

We update thermal ﬁelds using:

θn+1

p=θn

p+∆t∑

i

˙

θn+1

iSip , (62)

qn+1

p=qn

p+κp∆t∑

i

˙

θn+1

i∇xSip , (63)

where κpis the heat conductance at each particle p;∆tis the time increment; all superscripts n,n+1 refers

to the variable at the nth, n+1th time step.

When updating velocities, we use a combination of ﬂuid-implicit-particle (FLIP) and particle-in-cell

(PIC) velocity update scheme (Stomakhin et al.,2013):

vn+1

p= (1−ζ) vn

p+∆t∑

i

an+1

iSip !+ζ∑

i

vn+1

iSip (64)

where vi,vpare velocities at grid node ior particle p, respectively. ζis a coefﬁcient indicating the portion

of PIC update in the velocity evolution scheme. ζ=1 gives a pure PIC update of velocities, while ζ=0

gives a pure FLIP velocity update. Note that, because the PIC velocity update scheme can be regarded as a

spatial smoothing scheme on the velocity ﬁeld, it may damp out the high-frequency oscillations (Liu et al.,

2018). The locations and deformation gradients associated with the p−th material points are updated as

follows:

xn+1

p=xn

p+∆t∑

i

vn+1

iSip , (65)

Fn+1

p= I+∆t∑

i

vn+1

i⊗ ∇xSi p!Fn

p. (66)

12 Mian Xiao et al.

where xpand Fpare the location of particle center and the deformation gradient at p, respectively; Ide-

notes the 2nd-order identity tensor; ⊗is the dyadic product operator. The stress tensor and damage scalar

are updated upon obtaining the updated mechanical and thermal ﬁelds.

3.2 The damage ﬁeld-gradient material partitioning

To identify the locations of the contacts, we employ the partition criterion ﬁrst proposed in Homel and

Herbold (2017) where we would use the gradient of damage ﬁeld at the nodes of grid ias an indicator

function to partition the continuum bodies and identify the potential contacted interfaces. In essence, this

approach detects contacts by assuming that contacts may take place at damaged interfaces, and one may

detect contacts by identifying traits of the gradient of damage ﬁeld projected onto the background mesh to

identify the set of contact nodes and compute the relative separation between the contacted bodies. While

previous works such as Liu and Sun (2020a) and Kakouris and Triantafyllou (2019) may require one to

assigning labels or level set to a subset of material points to identify the bodies in contacts, the treatment

in Homel and Herbold (2017) bypasses this requirement and hence is ideal for simulating fragmentation

where self-contacts must be detected (de Vaucorbeil et al.,2019). With i-th sub-domain of bodies partitioned

via the node set i, we can elaborate on the relative velocity between different parts of the body to further

determine the contact interactions. The partitioning scheme is brieﬂy reviewed here for completeness.

The normalized smoothed damage ﬁeld is constructed as:

D(x) = D(x)

S(x), with D(x):=∑

p

max(sp,dp)ω(¯

r),S(x):=∑

p

ω(¯

r), (67)

where spis a boundary particle indicator such that sp=1 for boundary particles and sp=0 otherwise,

dpis the damage parameter at particle p,ω(¯

r) = 1−3¯

r2+2¯

r3for 0 ≤¯

r≤1 and ω(¯

r) = 0 elsewhere;

¯

r=||x−xp||/rpis a normalized distance measure with a support radius of rp. For plane 2D problems, rp

is usually chosen as the diagonal length of the background cells. The setting of spis adopted to consider

the self-contact (different bodies existing initially), and the setting of S(x)is employed to eliminate the

boundary effect.

The gradient of damage ∇xD(x)could help identify particles from different ﬁelds. Suppose there is

a developing crack crossing a grid node, which makes ﬁeld partitioning necessary at that node. Usually,

the degree of damage of a particle closer to the surface is much larger than that of internal particles, so

the damage gradients typically point from surfaces (material boundary) to internal regions. Therefore,

particles belonging to different material ﬁelds in the inﬂuencing region of a grid node generally have

damage gradients pointing in relatively opposite directions. To describe this mathematically, we claim the

following criterion for ﬁeld partition at grid nodes:

∃particles p1,p2in the vicinity of node is.t. ∇xDi· ∇xDp1>0 and ∇xDi· ∇xDp2<0. (68)

where ∇xDirefers to the gradient of Dat i, and ∇xDp=∇xD(xp). For (68) to be valid, it is necessary

that ∇xDi6=0, but around a fully damaged region, we probably get the value close to zero. Therefore,

a nonlocal approach is required to determine the grid-node damage gradient. Since we only compare the

sign of vector dot products rather than the magnitude and the order of numbering for material ﬁelds does

not matter, the following rule for determining the nonlocal damage gradient is adopted:

∇xDi=∇xD(xmax

i)where xmax

i=arg max

|x−xi|<rg

| ∇xD(x)|. (69)

where xiis the position of grid node i,rgis a support radius for the range of nonlocal searching, usually

chosen as the diagonal length of the grid cell, and | · | stands for the vector norm operation. To avoid

maximizing a complicated function in a continuous space, we calculate (69) by searching for the damage

gradient with the maximum vector norm over ∇xDplocated at particle centers. Once we divide the bodies

into different ﬁelds, the update scheme should be performed separately for different bodies.

Since we use an evolving damage ﬁeld to identify the location of the fractured interface, an impor-

tant issue is that there could be incompletely damaged region(s) that should not be regarded as actually

MPM Thermo-Contact 13

fractured zones during the fracture. To resolve this issue, we may determine the degree of damage at a

grid node. If a grid node is completely damaged (fractured), the two different material ﬁelds may sepa-

rate apart and we should apply the contact algorithm. Otherwise, cohesion should be allowed between

different ﬁelds, making the partitioned material ﬁelds still evolve as a single entity. To deﬁne a separable

condition that distinguishes the separable and non-separable state at a grid node, we adopt the average

nodal damage for two material ﬁelds to evaluate the state of damage (Homel and Herbold,2017):

Dki =∑kp mpdpSip

mki

, (70)

where the subscript kdenotes the k-body. The separable condition is: a grid node is separable if its maxi-

mum and minimum averaged nodal damage reaches some critical values, namely:

max(D1i,D2i)>Dcr, min(D1i,D2i)>Dmin , (71)

where Dcr indicates the critical averaged nodal damage required for separation for the maximum over D1i

and D2ito reach, and Dmin indicates the minimum averaged nodal damage required for separation for the

minimum over D1iand D2ito reach.

If the partitioned ﬁelds are separable from each other, we will further apply a thermo-contact algorithm

to determine all the contact interactions, especially the contact forces and contact thermal loads. For cases

where the partitioned ﬁelds are not separable, both ﬁelds evolve as a single one integrally.

3.3 Calculations of the contact force and heat ﬂux across contacts

In the MPM, a momentum correction scheme is widely used to approximate the contact forces at grid

nodes (Bardenhagen et al.,2001). We are speciﬁcally interested in a double-ﬁeld formulation of material

partitioning for contact modeling. To formulate the momentum conservation over both ﬁelds, we deﬁne

the center-of-mass velocity as an evaluation of the averaged velocity if merging both ﬁelds into one: (Bar-

denhagen et al.,2001;Nairn,2003):

vcm

i=∑2

kmki ˆvki

∑2

kmki

, (72)

where vcm

iis the nodal center-of-mass velocity, ˆvki represents the updated nodal velocity for the kth mate-

rial ﬁeld before contact adjustment.

A potential in-contact node is deﬁned as a grid node with particles from different material ﬁelds in its

vicinity. However, real contact interactions only take place when two material ﬁelds have a trend of inter-

penetration. We need the following criterion to distinguish if two material ﬁelds are coming into contact:

(ˆvki −vcm

i)·nki >0, (73)

where nki is the outward surface normal vector based on the nodal mass gradient (Homel and Herbold,

2017;Liu et al.,2018):

n1i:=1

|ˆn1i−ˆn2i|(ˆn1i−ˆn2i),n2i:=−n1iwith ˆnki =∑kp mp∇xSip

|∑kp mp∇xSi p |. (74)

Once contact is conﬁrmed at a grid node, we will perform the calculation of contact forces. The de-

termination of normal contact forces should address the normal KKT conditions by eliminating potential

inter-penetration. Subject to momentum conservation, the normal contact force fcont

ni at node iis calculated

as:

fcont

ni =1

∆tmki (vcm

i−ˆvki)·nki, (75)

One can verify that substituting kwith either 1 or 2 into (75) gives the same result. Following the same

logic of momentum correction, the tangential contact forces should be calculated as: fcont

ti =1

∆tmki (vcm

i−

14 Mian Xiao et al.

ˆvki)·tki, where tki is the right-handed contact tangential vector perpendicular to nki . However, the tan-

gential KKT conditions specify that the tangential contact forces cannot exceed µc|fcont

ni |. This requires the

following correction on the tangential contact forces:

fcont

ti =min{| ˆ

fcont

ti |,µc|fcont

ni |} · sign(ˆ

fcont

ti ),ˆ

fcont

ti :=1

∆tmki (vcm

i−ˆvki)·tki, (76)

Combining the normal and tangential components gives an expression for total the nodal contact forces:

fcont

ki =fcont

ni ·nki +fcont

ti ·tki . (77)

We now focus on deriving an expression for the contact thermal load term Qcont

ki . For the simplicity of

derivation, we assume that the power scale e=1 so that the simpliﬁed contact heatﬂux constitutive law

(13) is used, and ¯

γis uniform all over the contact boundary. We start by inserting (13) and (14) into (61):

Qcont

ki =ZΓc¯

γpcn[θ]k−1

2pct ˙

δtSidΓ, (78)

Extracting the temperature jump [θ]and the relative slip rate ˙

δtout of the integral in (78), we have:

Qcont

ki =¯

γ[θ]avg

ki ZΓc

pcnSidΓ−1

2(˙

δt)avg

ki ZΓc

pctSidΓ, (79)

where [θ]avg

ki and (˙

δt)avg

ki are the averaged temperature jump and slip rate observed at the kth material ﬁeld

at node i, w.r.t. the original integral in (78). For [θ]avg

ki , we can use the assembled nodal temperatures as a

practical approximation:

[θ]avg

ki ≈(θ2i−θ1i:= [θ]1ik=1,

θ1i−θ2i:= [θ]2ik=2. (80)

For contacts with smooth surfaces, we assume that the spatial variation of contact surface normal vector

is sufﬁciently low such that the following approximation for normal and tangential contact forces holds:

fcont

ni ≈ZΓc

pcnSidΓ,fcont

ti ≈ZΓc

pctSidΓ. (81)

Substituting (80) and (81) into (79) gives an expression for calculating the nodal thermal contact load

term:

Qcont

ki ≈¯

γ[θ]ki ·fcont

ni −1

2(˙

δt)avg

ki ·fcont

ti . (82)

Remark 1. Alternative formulation for the computation of the contact force. The focus of this work

is to simulate dynamic multi-body contact under a high loading rate in which case the contact model is

primarily designed for capturing the conservation of momentum during collision. To improve efﬁciency,

the prevention of interpenetration of the contacts in the dynamic MPM simulations is often enforced via a

collision contact algorithm that balances the momentum of the collided bodies, rather than directly enforc-

ing the interpenetration constraints, as we explained in this section (cf. Bardenhagen et al. (2001); Homel

and Herbold (2017); Han et al. (2019)).

For quasi-static contact mechanics problem, the collision MPM contact algorithm that updates the ve-

locity ﬁeld may not be suitable and hence an alternative formulation that accurately resolves the static

contact forces based on gap function is presented here for completeness. This alternative formulation is

used in the Hertzian contact numerical example (see Section 4.1). To resolve the contact precisely for the

quasi-static case where the velocity is negligible, one may calculate the gap function between the two inter-

faces and obtain the force required to prevent the interpenetration. An implicit MPM model dedicated to

quasi-static contact problems can be found in our previous work that generates the gap function via a level

set approach, i.e.,Liu and Sun (2020b). A comprehensive review on this subject can be found in Wriggers

and Zavarise (2004).

For the ease of implementation, we employ a penalty method to relate normal contact pressures from

gap function values:

pcn =κnδn(83)

MPM Thermo-Contact 15

where κnis a normal penalty parameter for interpenetration and δnis the normal component of the gap

function. In the tangential direction, the Coulomb friction law can be implemented in a manner similar

to an elasto-plastic material model. To describe this behavior, we decompose the tangential slip into an

”elastic” and a non-recoverable ”plastic” part: δt=δe

t+δp

t(Laursen,2013). The constitutive law for the

tangential frictional response is then stated as follows:

φ(pct,pcn):=|pct | − µc|pcn | ≤ 0,

˙

δp

t=˙

λpct

|pct|,˙

λ≥0, ˙

λφ =0. (84)

where φis termed the slip function (analogous to yield function in theories of plasticity) (Liu and Sun,

2020a), and ˙

λindicates the slip rate. The tangential contact traction is then calculated as:

pct =κt(δt−δp

t)(85)

where κtis the tangential penalty parameter. For frictional contacts, the traction on the contact boundary

is pc=pcnn+pct t. Insert this expression for pcinto (50):

fcont

i=ZΓc

Siκnδn·n+κt(δt−δp

t)·tdΓ(86)

For the practical calculation of fcont

iin (86), we adopt a the concept of boundary layer following Bandara

and Soga (2015), where we re-express (86) with the following manner:

fcont

i≈∑

p

sp1

lp

Sip κnδn p ·np+κt(δtp −δp

tp )·tp(87)

where δnp,δtp ,δp

tp are the normal gap, the total tangential gap, and the plastic part of the tangential gap

at particle p, respectively. npand tpindicate the normal and tangential directions at particle p.lpis the

thickness of particle pin the contact normal direction. The expression inside the summation is multiplied

by the boundary indicator spsuch that only the values at boundary particles are used to determine the

integrated nodal contact force.

3.4 Algorithms for MPM with evolving thermo-mechanical contacts

In this section, we elaborate on the time-stepping algorithm and provide an algorithmic overview for

the implementation with MPM. In each time step, we ﬁrst construct a particle-to-grid projection and obtain

the smoothed damage-gradient ﬁeld. Next, we introduce a thermal-mechanical sub-stepping algorithm

where an isothermal split enables us to solve the thermo-mechanical problems by updating the tempera-

ture and displacement ﬁeld sequentially (Simo and Miehe,1992;Wriggers et al.,1992;Wollny et al.,2017;

Suh et al.,2020;Suh and Sun,2019). The sequential solver ﬁrst updated the temperature (and heat ﬂux)

ﬁeld by solving the balance of energy with a ﬁxed displacement ﬁeld. Following the updates of the ther-

mal ﬁelds, the mechanical ﬁelds (displacement and damage) are updated by solving the balance of linear

momentum equation. Like other operator-splitting schemes, this sequential approach may lead to consis-

tency issues as such the time step must be sufﬁciently small to avoid departure from the equilibrium states.

However, since the upper bound for the stable time step for the explicit time integrator is usually small,

this issue is usually not signiﬁcant (Tao et al.,2018).

16 Mian Xiao et al.

Algorithm 1 Sequential solver for thermo-mechanical problems with evolving contacts

1: Calculate the weight functions Sip and the corresponding gradients ∇xSi p between material points

and grid nodes.

2: Calculate the smoothed damage ﬁeld Dpand the corresponding gradient ∇xDpfor all particles.

3: Find the damage gradient ∇xDifor all nodes.

4: For each node, determine if it is cracking or undergoing contact with (68).

5: Evolve the thermal ﬁelds for all particles. See Algorithm 2.

6: Evolve the kinematic and mechanical ﬁelds for all particles. See Algorithm 3.

The overall explicit algorithm is listed in Algorithm 1. The detailed time integration algorithms for

individual thermal or mechanical ﬁelds are provided in Algorithms 2and 3respectively.

Algorithm 2 Incremental updates for the MPM thermal ﬁelds

1: Calculate the nodal temperature evolution rate:

2: for all node ido

3: if Eq (68) holds true then

4: Calculate Cki for the k-th material ﬁeld via (56).

5: Calculate Qext

ki ,Qint

ki ,Qecpl

ki ,Qdcpl

ki for the k-th material ﬁeld via (57-60).

6: Find the total nodal thermal load without contact adjustment: ˆ

Qki =Qext

ki +Qint

ki +Qecpl

ki +Qdcpl

ki .

7: if (71) holds true then

8: Calculate Qcont

ki from (82) using the mechanical results from the last time step.

9: else

10: Obtain Qcont

ki such that ˙

θ1i=˙

θ2i=ˆ

Q1i+ˆ

Q2i

C1i+C2iafter adjustment on the contact thermal loads.

11: Adjust the rate of temperature change: ˙

θki =ˆ

Qki +Qcont

ki

Cki .

12: else

13: Calculate single-ﬁeld Ci,Qext

ki ,Qint

ki ,Qecpl

ki ,Qdcpl

ki via (56)˜(60) (where k=1 only)

14: Calculate the single-ﬁeld ˙

θki (k=1) with (55).

15: Update all particle temperatures via (62).

16: Update the nodal temperature ﬁeld using the updated particle temperatures:

17: for all node ido

18: if (68) holds true then

19: Calculate the nodal temperature at different ﬁelds: θn+1

ki =1

Cki ∑kp mpcpθn+1

pSip

20: else

21: Update the single-ﬁeld nodal temperature: θn+1

i=1

Ci∑pmpcpθn+1

pSip

22: Update all particle heat ﬂuxes with (63).

MPM Thermo-Contact 17

Algorithm 3 Incremental updates for the MPM mechanical ﬁelds

1: Calculate the nodal acceleration:

2: for all node ido

3: if (68) holds true then

4: Calculate mki with (47) and Dki via (70) for the kth material ﬁeld.

5: Calculate the nodal velocities for both material ﬁelds with: mki vn

ki =∑kp mpvn

pSip .

6: Find the contact normal vectors nki via (74).

7: Calculate fext

ki and fint

ki via (48) and (49).

8: Update ˆvki without contact adjustment: ˆvn+1

ki =vn

ki +∆tfext

ki −fint

ki

mki

9: if (71) holds true then

10: Calculate the center-of-mass velocity vcm

ifrom (72).

11: if n1i·(ˆvn+1

1i−vcm

i)>0then

12: Find the frictional coefﬁcient with (15) using the updated temperatures θ1i,θ2i.

13: Calculate the contact force fcont

ki via (77).

14: else

15: Set fcont

ki as zero vectors.

16: else

17: Obtain fcont

ki such that vn+1

1i=vn+1

2i=vcm

iafter contact adjustment on nodal velocities.

18: Adjust the nodal velocities with: vn+1

ki =ˆvn+1

ki +fcont

ki

mki ∆t

19: Find the nodal accelerations for both ﬁelds with (46).

20: else

21: Calculate the single-ﬁeld nodal attributes mki ,fext

ki ,fint

ki (where k=1 only).

22: Calculate the single-ﬁeld nodal velocities with: mki vn

ki =∑pmpvn

pSip (where k=1 only).

23: Calculate the nodal acceleration (46) (where k=1 only).

24: Update the nodal velocity with: vn+1

ki =vn

ki +∆t·at

ki (where k=1 only).

25: Update the particle locations, velocities, and deformation gradients via (65-66).

26: Update the particle stresses with the corresponding constitutive law, using the updated mechanical

and thermal results.

27: Evolve the particle damage via Rankine’s damage law.

4 Numerical Examples

This section presents several numerical examples to verify and validate the implementation of the

MPM contact models and demonstrate the capacity of the proposed model in handling evolving multi-

body thermo-mechanical contacts due to fracture and damage. We ﬁrst verify the implementation of the

MPM contact models with a Hertzian contact problem and a thermal contact problem. Simulation results

of these two examples are compared with analytical solutions and in the ﬁrst example, a mesh reﬁnement

study has been conducted. Furthermore, a validation exercise against the Kalthoff-Winkler dynamic frac-

ture experiment is included to test whether the Rankine damage model implemented in the MPM frame-

work may replicate the same crack pattern observed in experiments. We then demonstrate the capacity of

the model to simulate thermal-mechanical damage and contact mechanics for multiple bodies in contact. A

disk-squish problem is introduced to demonstreate the MPM model’s capacity to solve multi-body thermo-

mechanical contact problems, while a three-grain particle fragmentation problem is used to demonstrate

the capability to simulate evolving thermo-mechanical contacts during the fracture and subsequent frag-

mentation process. We then analyze how impact velocity affects the damage, fracture, and fragmentation

of particle assemblies during the non-isothermal fragmentation process.

18 Mian Xiao et al.

4.1 Veriﬁcation against Hertz’s contact benchmark problem

This numerical example is included to verify the implementation of the MPM simulations by compar-

ing the analytical solution for a Hertz contact problem with non-ﬂat contact surfaces. We replicate the same

simulation conﬁguration previously used in Liu and Sun (2020a) while the height of the body is changed

to 4mm, as shown in Fig. 1. Body 1 exhibits a 10GPa Young’s modulus of 10 GPa and 0 Poisson ratio, while

Body 2 is considered rigid. The corresponding analytical solutions for the radius of the contact area band

the maximum contact pressure pmax are listed in Eq. (88) (cf. Hertz (1882); Johnson (1982); Barber (2018)),

b=2r2FR

πE0,pmax =2F

πb(88)

where Fis the magnitude normal line load, R=R1R2

R1+R2is the equivalent body radius, where R1,R2are the

radii of the two contact surfaces, respectively. As the contact surface on the lower body is ﬂat, its radius

is inﬁnite and Rcan be simpliﬁed as the radius of the upper contact surface. E0is the effective Young’s

modulus deﬁned as: 2

E0=1−ν2

1

E1+1−ν2

2

E2where E1,ν1and E2,ν2are Young’s modulus and Poisson ratio for

the two contacting elastic bodies, respectively. With b,pmax deﬁned, the distribution of contact pressure

along the contact surface can be expressed as follows:

pcn =−pmaxr1−(s

b)2for 0 ≤s≤b(89)

where pcn is the contact pressure (negative in sign) and sis the distance to the center of the contact area.

Since our MPM time integration is explicit, the static equilibrium solution is obtained via a dynamic relax-

ation method (Tu and Andrade,2008;Liu et al.,2016). The vertical force applied on the top boundary (per

unit width) is 156.7 N/mm. This value is substituted into (88) to determine the analytical solution on the

contact surface.

We perform a mesh convergence study on this problem by discretizing the domain into three mesh

sizes (size of the cells in the background grid): 0.2 mm, 0.1 mm, and 0.05 mm. The density of material

points in a cell is always 4 (as a 2 ×2 grid) except for some cells located near the domain boundary.

Fig. 1: Conﬁguration for the Hertz problem (unit in the ﬁgure is in mm).

We collect the contact forces on the grid nodes and use them to compute the normal contact pressure.

We then compare the simulation results with the analytical solution of the normal reaction loading in Fig. 2.

MPM Thermo-Contact 19

We observe that, the simulated contact pressure converges into the analytical pressure distribution upon

consecutive mesh reﬁnement, as indicated by the root mean square error (RMSE) of the pressure over

collocation points at the contact boundary. The RMSE normalized by the maximum contact pressure is

shown in Fig. 3. The stress distributions of all three cases are presented in Fig. 4, where we clearly observe

a stress concentration near the contact boundary, and the maximum stress value is close to the maximum

analytical contact pressure. These ﬁndings verify our approach to resolve contact problems with a gap

function.

Fig. 2: Comparison of simulated contact pressure and Hertzian analytical contact pressure distribution on

the contact surface. (a) mesh size 0.2 mm, (b) mesh size 0.1 mm, (c) mesh size 0.05 mm

Fig. 3: Convergence of relative RMSE on the simulated contact pressures with different mesh sizes.

4.2 Veriﬁcation against Wrigger’s thermal contact benchmark problem

The purpose of this numerical example is to verify the thermo-mechanical contact MPM simulation

between two contacting squared blocks subjected to both mechanical loading and thermal gradient. Pre-

viously, the same boundary value problem has been used to verify a ﬁnite element model for thermo-

mechanical contact in Wriggers and Miehe (1994). The boundary conditions and the domain of this initial

boundary value problem are shown in Fig. 5(a). The length lfor both blocks is 2.5 mm and the background

grid cell size is 0.5 mm. The density of material point in a cell is 4 everywhere. We ﬁx the temperature on

the top and bottom as θhot =50◦C,θcool =20◦C. A uniform pressure pis applied on the top to ensure that

20 Mian Xiao et al.

Fig. 4: Convergence study for the Hertzian contact problem with the stress contour displayed on material

point clouds. (a) mesh size 0.2 mm, (b) mesh size 0.1 mm, (c) mesh size 0.05 mm.

there is a heat transfer across the interface due to contact pressure. The motion of the lower body at the

bottom is ﬁxed. We use the same material properties for both blocks speciﬁed in Table 1, which are typical

aluminum properties. The thermal expansion is ignored. This example is simulated using ∆t=5×10−8s,

and we specify ζ=1 so that a steady-state solution can be obtained.

(a) (b)

= 50°C

= 20°C

Fig. 5: Conﬁguration of the contact heat conduction problem: (a) geometry and boundary conditions, (b)

vertical proﬁle of the steady-state temperature.

The analytical solution of the temperature ﬁeld along the vertical axis is piece-wise linear with a jump

at the contact surface, with the speciﬁc temperature θ1and θ2for the upper and lower body on the contact

surface (cf. (Wriggers and Miehe,1994)):

θ1=(1+κc)θhot +κcθcool

1+2κc,θ2=(1+κc)θcool +κcθhot

1+2κc,κc:=hcl

κ, (90)

where hcis calculated from the normal contact stress (which is equal to p) with (11). As for the surface con-

ductance coefﬁcient ¯

γ, we assign it with a rather large value( equals to 1) to accelerate the convergence to

MPM Thermo-Contact 21

Table 1: Material properties (Aluminum) for the contact heat transfer veriﬁcation

Young’s modulus (GPa) E70

Poisson’s ratio ν0.33

density (g/cm3)ρ2.7

speciﬁc heat (m2/(s2K)) c900

conductivity (N/(sK)) K150

0 20 40 60 80 100 120

20

25

30

35

40

45

50

Fig. 6: Temperature at the contact surface versus the contact pressure.

the steady state. The comparison between analytical results and the output from our numerical simulation

is provided in Fig. 6. We observe that the stationary temperature on the two sides of the contact boundary

obtained from the MPM simulation matches the analytical solutions well. This numerical example suggests

that the proposed MPM is capable of simulating thermal-mechanical contacts with pressure-dependent

conductance.

4.3 Validation exercise against Kalthoff-Winkler dynamic fracture experiment

The purpose of this numerical example is to validate the MPM model via an experiment reported by

Kalthoff and Winkler where an edge-cracked metal plate is impacted by a projectile (Kalthoff and Winkler,

1988), which persists as a veriﬁcation for the crack modeling with damage ﬁeld in this paper. Due to the

embedded symmetry of this experiment, we only model the upper half of the plate, where we set the

boundary conditions at the bottom as symmetric boundary conditions. To simplify the impact loading, we

assume the projectile has the same elastic impedance as the plate so that we can apply half of the impact

rate as a velocity boundary condition to the surface being impacted (Song et al.,2008). The conﬁguration

is shown in Fig 7. The background grid cell size is 1 mm. The cell density of material point is 4 per cell

everywhere. We apply the impact load on the specimen with two velocities: 33m/s and 100m/s, where v0

should be set as 16.5m/s and 50m/s as explained before.

The specimen is composed of steel. To utilize the Rankine damage model, we specify the critical crack

energy and the critical failure stress as Gf=22.13N/mm and σf=570MPa, respectively (Homel and

Herbold,2017).

Fig. 8shows the results of crack propagation with applied impact velocity v0=16.5m/s, which cor-

responds to the standard impact velocity in Kalthoff and Winkler ’s experiments. The fracture pattern is

22 Mian Xiao et al.

25 mm

50 mm

100 mm

100 mm

Fig. 7: conﬁguration of the impact fracture problem

Table 2: Material properties of the steel that composes the specimen used in the dynamic fracture experi-

ment.

Young’s modulus (GPa) E190

Poisson’s ratio ν0.3

density (g/cm3)ρ7.8

speciﬁc heat (m2/(s2K)) c460

conductivity (N/(sK)) K55

thermal expansion (1/K) α1.0 ×10−5

failure stress (MPa) σf570

fracture energy (N/mm) Gf22.13

indicated by the damage ﬁeld. But according to the separation criterion in (71), fracture separation is trig-

gered after all particles in the vicinity of a node are ”fully damaged”. In order to correctly retrieve the

fracture propagation from such nodal information, it only matters where the damages at all associated

particles are very close to 1. In this sense, we regularized the dimensionless scalar damage exponentially

by dreg = (e5d−1)/(e5−1). As shown in Fig. 8, we successfully capture a crack propagating along the

direction with an angle around 67◦starting at the end of the pre-notched crack, and this corresponds to the

experimental results.

Apart from the standard veriﬁcation problem, we are also interested in illustrating the capability of

capturing complicated fracture patterns induced by fracture branching under a higher loading rate. Hence,

we apply an impact velocity v0=50m/s and compare the simulation results with another numerical study

of fracture branching in this impact fracture test. We observe that the resulting fracture pattern branches

on the major tensile crack.

In addition to replicating crack propagation, we also investigate on the thermal response in this dy-

namic fracture problem. The temperature ﬁelds of both cases are shown in Fig. 10 and Fig. 11. We observe

that there is a noticeable temperature increase at the lower corner of the pre-notched cracked, which is

caused by stress concentration at this location and strong coupling effects. Interestingly, the crack tip cools

down while the crack path behind the tip heats up. These observations are consistent with experimental

and numerical ﬁndings reported in Bougaut and Rittel (2001) in which crack tip cooling has been observed

due to the thermo-elastic cooling effect.

MPM Thermo-Contact 23

(a) t=41.34µs(b) t=53.27µs

Fig. 8: Damage ﬁeld at different times when impacted at 33m/s

(a) t=27µs(b) t=45µs

Fig. 9: Damage ﬁeld at different times when impacted at 100m/s

4.4 Thermomechanical coupling effect in dish-squish problem

In this numerical example, we adopt the ”disk-squish” boundary value problem originally proposed

in Homel and Herbold (2017) and incorporate the thermo-mechanical coupling effect to investigate the

frictional heat generated via contacts and how the friction-induced heat affects the deformation and contact

evolution in return in this multi-body contact problem. The conﬁguration is shown in Fig. 12. The cell size

of the background mesh is 1 mm. The cell density of material point is always 4 per cell except for those cells

adjacent to the boundaries of solid bodies. There are 2905 material points in total. The material properties

are listed in Table 3, which applies to all three distinct bodies. The initial frictional contact coefﬁcient is set

as 0.4. The horizontal component of the prescribed traction pshown in Figure 12 is 100MPa. We use a time

step of 2 ×10−8s.

We perform a series of simulations for three different conditions: (1) isothermal conditions; (2) thermo-

mechanical coupling without the thermo-mechanical softening effects on the frictional coefﬁcient µc; and

(3) thermo-mechanical coupling with softening on the frictional coefﬁcient µc, where θdam =800◦C and

24 Mian Xiao et al.

(a) t=41.34µs(b) t=53.27µs

Fig. 10: Temperature ﬁeld at different times when impacted at 33m/s

(a) t=27µs(b) t=45µs

Fig. 11: Temperature ﬁeld at different times when impacted at 100m/s

θref =20◦C. Fig. 13 and Fig. 14 show the results of both kinematic ﬁelds and thermal ﬁelds. In fact, the

frictional coefﬁcient µc=0.4 is sufﬁciently large that the body motions and xvelocity distribution under

both isothermal and thermo-mechanical coupling conditions are close to the non-slip contact response in

Homel and Herbold (2017), as shown in Fig. 13 (a), (b), (c) and (d).

Comparing Fig. 13 (a) and (b) with Fig. 13 (c) and (d) reveals that introducing the thermo-mechanical

coupling effects alone does not affect the ﬁnal conﬁguration signiﬁcantly. The difference in the motions

and velocity distributions of these two cases is minor. However, as shown in Fig. 14 (b), the contact surface

can heat up to a temperature over 900◦C due to friction and structural heating if the thermo-mechanical

coupling effects are considered. If the constitutive responses of the interfaces are thermally sensitive, then

the frictional heating may produce signiﬁcant enough changes in the mechanical responses. This scenario

is exhibited in the last simulation where the thermal softening of the frictional coefﬁcient µcis considered.

Fig. 13 (e) and (f) demonstrate the motions and velocities of the three bodies under the conditions where

µcdecreases as temperature rises. Due to the loss of friction caused by the increased temperature, the disk

in the last simulation got squeezed out of the sidewall of the rectangular body and hence the velocity of

MPM Thermo-Contact 25

15 mm

30 mm

3.5 mm

7 mm

15 mm

14 mm

21 mm 9 mm

Fig. 12: Conﬁguration of the disk squish problem

Table 3: Material parameters for the disk-squish test

Young’s modulus (GPa) E2.25

Poisson’s ratio ν0.125

density (g/cm3)ρ1.0

speciﬁc heat (m2/(s2K)) c500

conductivity (N/(sK)) K50

thermal expansion (1/K) α1.0 ×10−5

the disk is now noticeably higher in Fig. 13 (f) than those exhibited in Fig. 13 (b) and (d) where the friction

on the contact is sufﬁcient to prevent the sliding of the disk. These numerical simulations indicate the

importance of incorporating the two-way coupling thermal-mechanical effect on the constitutive responses

for the frictional contact.

As for the thermal responses, the thermal-induced reduction on frictional coefﬁcient does lead to less

heat generated at the contact. This behavior is attributed both to the difference in constitutive responses as

well as the difference in the deformed geometry of the contacts affected by the constitutive responses and

the two-way thermal-mechanical couplings as demonstrated in Fig. 14. Another interesting effect worth

noticing is that the simulations are conducted near the adiabatic limit such that the dominated heat transfer

mechanism is the convection due to the movement of the bodies. As indicated in Fig. 14, the temperature

rise only takes place in few particles around the frictional contact surface, but it does not diffuse into

the interior region of the bodies. Although the material is in the convection-dominated regime, the MPM

framework does not trigger any spurious oscillation due to the usage of the Lagrangian description of

motion for the material points.

26 Mian Xiao et al.

(a) isothermal condition, t=86µs(b) isothermal condition, t=106µs

(c) thermo-coupled condition, t=86µs(d) thermo-coupled condition, t=106µs

(e) thermo-coupled condition with damage on fric-

tion, t=86µs

(f) thermo coupled condition with damage on fric-

tion, t=106µs

Fig. 13: The horizontal component of the velocity ﬁeld at different time steps for the isothermal case (a and

b), the thermo-mechanical case without thermal-dependent friction (c and d), and the thermo-mechanical

case with thermal-dependent friction (e and f).

MPM Thermo-Contact 27

(a) thermo-coupled condition, t=86µs(b) thermo-coupled condition, t=106µs

(c) thermo-coupled condition with damage on fric-

tion, t=86µs

(d) thermo-coupled condition with damage on fric-

tion, t=106µs

Fig. 14: temperature distribution at different time steps for the thermal-mechanical case without thermal-

dependent friction (a and b) and with thermal-dependent friction (c and d).

28 Mian Xiao et al.

4.5 The three-grain fragmentation problem

In this last example, we conduct a set of grains crushing simulations in a two-dimensional domain

where an impact load is applied on the top of a column of three particles lumped on top of each other. This

boundary value problem mimics the experiments reported in Cil and Alshibli (2012) but is not a direct

digital replica due to the 2D idealization. Capturing the real experiments in large-scale three-dimensional

simulations may require signiﬁcant improvement on computational efﬁciency with GPU-enabled parallel

computing. Such an extension will be considered in future studies but is out of the scope of this work.

In this last set of numerical experiments, we conduct three simulations in which the top rigid platten are

prescribed with three velocities: 8m/s, 16m/s, and 80m/s, with a time step of 10−8s. The simulation con-

ﬁguration presented by the material point cloud is shown in Fig 15. The cell size is 0.1 mm and the density

of material points in a cell is always 4 (as a 2 ×2 grid) except for some cells located near the domain bound-

ary. There are 37627 material points in total. Meanwhile, the sidewalls and the bottom platten are rigid and

ﬁxed without any movement. The material properties of the three grains are identical and homogeneous.

They are also typical for quartz sand (see Table 4).

Fig. 15: Conﬁguration for the three-grain fragmentation problem.

Young’s modulus (GPa) E50

Poisson’s ratio ν0.2

density (g/cm3)ρ2.3

speciﬁc heat (m2/(s2K)) c2000

conductivity (N/(sK)) K5

thermal expansion (1/K) α1.0 ×10−5

failure stress (MPa) σf25

fracture energy (N/mm) Gf0.1

Frictional coefﬁcient at 293K µco 0.2

Table 4: Material properties typical of rock for particle crash simulation

MPM Thermo-Contact 29

The effects of the loading rate on the global responses can be seen in the force-displacement curve

of Fig 16. As expected, the higher loading rate triggered a larger reaction force while the slower loading

rate leads to a lower peak reaction force. Nevertheless, since the granular assemblies only consist of three

particles initially, an interpretation based on homogenization is not appropriate. Furthermore, due to the

dynamic nature of the simulations, wave propagation within and across the particles may affect the reac-

tion force and hence the reaction force exhibits oscillation in the temporal domain. The detailed analysis

on the simulated fragmentation process, the effect of the loading rate on the evolving contacts and the

thermo-mechanical responses, as well as the role of the frictional heating on the fragmentation patterns are

provided in Sections 4.5.1 and 4.5.2 accordingly.

Fig. 16: Force - displacement response on the top loading platten

4.5.1 Morphology of fragmented particles at different loading rates

As the top grain is fragmented signiﬁcantly, the evolution of the topology and geometry of contacts

may signiﬁcantly affect the reaction force exerted on the top platten. As such, we analyze the morphology

of the fragmented particles using ImageJ for all three loading cases in order to examine how impact ve-

locity affects the dynamic fracture process. The morphological study is enabled by an open-source image

analysis software called ImageJ (cf. Abr`

amoff et al. (2004)). We mask material points with d=1 so that

individual fragments can be distinguished from the fully crushed zones near the crack surface. For each

of the three simulations of different loading rates, we analyze three snapshots of fragmentation patterns

taken at selected time steps during the fragmentation process: initially cracked, fragmenting, fully crushed.

Our focus is on analyzing the probability distribution of two important geometric measures, the equiv-

alent diameter (see Fig. 17) , the diameter of a sphere (circle in 2D) that shares the same area of the fragment

and the roundness (see Fig. 18) , a normalized shape descriptor that measures how close the particle cross

section resembles a circle, with a value of 1 indicates a perfect circle and zero indicates a line (Abr`

amoff

et al.,2004).

Fig. 17 shows the empirical cumulative distribution of the equivalent diameters for all three cases.

This data is obtained by ﬁrst identify all individual particles by identifying the boundaries of particles

through image segmentation (see Figs. 19,20, and 21). In all three cases, the impact-induced fragmentation

progressively reduces the mean equivalent diameter and increases the variance of the particle size. The

maximum equivalent diameter also decreases during fragmentation. These trends can be veriﬁed by the

fragment pattern shown in Figs. 19,20, and 21.

30 Mian Xiao et al.

Here the top grain is initially in one piece with some initial cracks and small comminuted particles

around. As the fragmentation progresses, the top grain is fragmented into smaller pieces of a variety of

sizes, while the number of fragments increases signiﬁcantly. This increase in the variety on particle size

explains why this empirical cumulative distribution is smoother and more distributed as the platen moves

downward. Fig. 17 also reveals that the high-speed impact tends to generate fragmentation of more dis-

tributed particle sizes. This trend may be attributed to the fact that the higher impact speed may promote

crack branching. As a higher external power is supplied to the granular assembles in the case with a higher

loading rate, crack branching may occur more frequently when the crack velocity reaches a threshold value

(Congleton and Fetch,1967). As the particles split, the equivalent diameter of the new fragmented parti-

cles gets smaller than that of their ”parent” particle and that further reduces the feasible length of the new

crack path in the fragmented particles. Consequently, this size effect of fragmentation lead to the particle

distribution becomes increasingly well graded as the impact load is applied.

(a) loading rate 8m/s (b) loading rate 16m/s (c) loading rate 80m/s

Fig. 17: Cumulative distribution of the equivalent diameter of fragmented particles (of the top particle)

for simulations with loading rates = (a) 8m/s, (b) 16m/s and (c) 80m/s. The blue, red and yellow curves

are obtained when the prescribed displacement on the top reaches 0.2mm, 0.8mm and 4mm accordingly

(except (c), see Table 4).

In addition to the size of fragments, we analyze the shape of fragments as well. The empirical distri-

butions of the fragment roundness for simulations with different loading rates are shown in Fig. 18. In all

three cases, the value of roundness of the fragmented particles tends to be smaller than those at the initial

or fully crushed stages. This suggests that when the top grain is being fragmented starting from the initially

cracked status, the newly generated fragments tend to be less rounded since the fragmentation may lead

the fragmented particles with a higher aspect than the intact ”parent” particle, a phenomenon also reported

in Zhao et al. (2015). However, when the fragmented particles are getting crushed further, the fragments

are more rounded. This can be explained by the fact that stress concentration is more likely to occur when

a sharp object is in contact with a smooth surface, which increases the probability of breakage into smaller

and more rounded pieces for those fragments. This observation is consistent with the Weibull theory for

the particle crushing process, which indicates that the survival probability of a fragmented particle under

crushing is a function of the nominal tensile strength (Voo,2000;Weibull et al.,1951).

4.5.2 Effect of frictional and structural heating on the fragmentation

One important reason that we incorporate thermo-coupling into the mechanical formulation is to study

the thermal effects involved during the fracturing process. To examine how thermo-mechanical coupling

affects the fragmentation process, we introduce a control experiment where the material is assumed to

remain under the isothermal condition and set the loading rate as 16m/s. We then compare the crack

pattern of the isothermal case and the thermo-mechanical coupling case at displacement = 16mm in Fig. 22,

where we masked fully damaged material points (d=1).

MPM Thermo-Contact 31

(a) loading rate 8m/s (b) loading rate 16m/s (c) loading rate 80m/s

Fig. 18: Cumulative distribution of the roundness of fragmented particles (of the top particle) for simula-

tions with loading rates = (a) 8m/s, (b) 16m/s and (c) 80m/s. The blue, red and yellow curves are obtained

when the prescribed displacement on the top reaches 0.2mm, 0.8mm and 4mm accordingly (except (c), see

Table 4).

(a) loading rate 8m/s (b) loading rate 16m/s (c) loading rate 80m/s

Fig. 19: Fragment patterns in the initial cracking stage where the prescribed displacement on the top=

0.2mm for (a) and (b) and 0.4mm for (c).

(a) loading rate 8m/s (b) loading rate 16m/s (c) loading rate 80m/s

Fig. 20: Fragment patterns after the fragmentation is triggered where the prescribed displacement on the

top= 0.8mm

Although the fragmentation pattern of the top grain looks similar, the crack pattern for the middle

grain and bottom the bottom grain differs from each other signiﬁcantly: in the isothermal case we observe

one major vertical crack in the middle grain, but it moves to the bottom grain in the thermo-mechanical

32 Mian Xiao et al.

(a) loading rate 8m/s (b) loading rate 16m/s (c) loading rate 80m/s

Fig. 21: Fragment patterns when the top grain is almost fully crushed where the prescribed displacement

on the top= 4mm

(a) isothermal conditions (b) thermo-coupling conditions

Fig. 22: Comparison of crack pattern between grain crushing simulations (a) without and (b) incorporating

thermo-mechanical coupling effect. The loading rate of the top platen is 16m/s.

coupling case. These results indicate that crack pattern is often sensitive to perturbation and hence even

the thermomechanical coupling effect with moderate temperature increase is capable of triggering a pro-

foundly different deformed conﬁguration. As the increased temperature reduces the frictional coefﬁcient,

and the contact stresses affect the thermal conductance of the contact, the evolution of contacts could be

profoundly altered by the heat transfer and vice versa.

Note that discrete element models or level set based splitting method that uses the homogenized stress

and temperature of each particle as the fragmentation criterion are incapable of replicating the interplay

among the frictional contact, the heat transfer inside and across the particle, and the stress concentration

at the particle contacts and hence may provide unrealistic reasons. This simpliﬁcation of mechanics and

geometry may have a profound effect on the interpretation of the energy scaling for the dissipated energy

after the fragmentation (Baˇ

zant,1993;Carpinteri and Pugno,2005). Further investigations on the implica-

tion of the energy scaling using the proposed MPM framework will be considered in future studies.

Figures 23-25 shows the temperature distribution in the deformed conﬁguration for the three simu-

lations with different loading rates. In Fig. 23, there are a few major cracks that split the top and middle

particles. The crack branching around these dominating cracks is very limited. In contrast, Fig. 24 shows

MPM Thermo-Contact 33

(a) displacement = 0.12mm (b) displacement = 0.4mm (c) displacement = 0.8mm (d) displacement = 4mm

Fig. 23: Temperature ﬁeld in the deformed conﬁguration of the particles under an impact velocity of 8m/s

on the top platen.

(a) displacement = 0.12mm (b) displacement = 0.4mm (c) displacement = 0.8mm (d) displacement = 4mm

Fig. 24: Temperature ﬁeld in the deformed conﬁguration of the particles under an impact velocity of 16m/s

on the top platen.

34 Mian Xiao et al.

that the crack branching is more profound in the top grain. Those branching and diverging cracks even-

tually fragment the top grain into smaller pieces but there is no crack propagated into the grains at the

middle and the bottom. In the last case where the impact velocity increased to 80m/s, the region with in-

creased temperature grows spatially due to thermal convection and no signiﬁcant crack branching occur

until a large portion of the prescribed displacement applied.

(a) displacement = 0.12mm (b) displacement = 0.4mm (c) displacement = 0.8mm (d) displacement = 4mm

Fig. 25: Temperature ﬁeld in the deformed conﬁguration of the particles under an impact velocity of 80m/s

on the top platen.

In all three cases, the thermal diffusivity is sufﬁciently low (relative to the loading rate) such that the

thermal convection and the local heat caused by the damage and frictional dissipation are factors that

dominate the temperature proﬁles. While the temperature rise in these simulations is not signiﬁcant, the

temperature distribution may nevertheless indicate the propagation of damage. By comparing Fig. 23,

Fig. 24 and Fig. 25, one may observe that the increase of the loading rate may promote crack branching

and allow the heat to accumulate locally without noticeable diffusion. The crack branching may lead to a

larger amount of energy dissipation at the top grain before a sufﬁcient amount of strain energy ﬂux causes

damage in the grains at the middle and the bottom.

Note that the material could have melted if (1) the speciﬁc heat is low and/or (2) the energy dissipa-

tion is large such that the local temperature may rise without signiﬁcant diffusion (Goldsby and Tullis,

2011;Ma and Sun,2020). Furthermore, a more profound temperature increase may also trigger the brittle-

ductile transition that affects the mechanical responses and fracture patterns (Choo and Sun,2018). These

mechanisms are not captured in this research but will be considered in future studies.

5 Concluding remarks

We propose a material point modeling framework designed to replicate the fracture and contact me-

chanics under non-isothermal conditions. We incorporate a smoothed damage ﬁeld gradient approach to

successfully identify the ﬁeld separation at potential contacting nodes while capturing the convection-

diffusion of heat with Lagrangian material points. To overcome mesh sensitivity, thermal-sensitive contact

MPM Thermo-Contact 35

and non-local damage models are used such that the degradation of the bulk material, as well as the ther-

mal softening of the frictional interface, can both be triggered without spurious mesh dependence. We

validate the resultant numerical schemes as well as use the resultant numerical models to simulate frac-

ture and fragmentation of particle assembles with evolving contacts. Our numerical results indicate that

the proposed model is capable of capturing the complex fracture patterns and the resultant contacts un-

der different strain rates. The role of the frictional heating on the crack growth, crack branching and the

resultant fragmentation processes are analyzed.

6 Acknowledgments

WCS is supported by the Dynamic Materials and Interactions Program from the Air Force Ofﬁce of

Scientiﬁc Research under grant contracts FA9550-17-1-0169 and FA9550-19-1-0318, and the Earth Materi-

als and Processes program from the US Army Research Ofﬁce under grant contract W911NF-18-2-0306.

MX is supported by the Presidential Fellowship of Columbia University. CL’s involvement is partially

supported by FA9550-19-1-0318 during his tenure as a research scientist at Columbia University. These

supports are gratefully acknowledged. The views and conclusions contained in this document are those of

the authors, and should not be interpreted as representing the ofﬁcial policies, either expressed or implied,

of the sponsors, including the Army Research Laboratory or the U.S. Government. The U.S. Government is

authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright

notation herein.

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