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DP-MPM: Domain partitioning material point method for evolving multi-body thermal-mechanical contacts during dynamic fracture and fragmentation

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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 implicit 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 thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems.
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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) = (1d)(U(J) + W(biso )), (2)
ρoM(J,d,θ) = 3(1d)αt(θθ0)U
J, (3)
ψt(θ) = c[(θθ0)θlog(θ/θ0)], (4)
respectively, where ρois the mass density on the unit reference volume, biso =J2/3brepresents 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 (1d), i.e.,
ψe(b,d) = (1d)ψe(b),M(J,d,θ) = (1d)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(θθ0)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= (1d)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 =(vt2vt1, for k=1,
vt1vt2, 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=(θ2θ1, for k=1,
θ1θ2, 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=κxθ, (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) = σmR(Cervera et al.,1995), where Ris the damage
internal variable. Φand Rshould satisfy the KKT condition, which is addressed as:
Φ0, ˙
R0, 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)(1σf
R),σf<R<σf(1+1
Hs),
1 , Rσf(1+1
Hs).(20)
where Hs:=Hslch/(1Hslc 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
ρoψe3αt(θθ0)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=θ2M
∂θ2θ2ψt
∂θ2, (33)
He=θ2M
∂θb+2ψt
∂θb:˙
b, (34)
Hin =θ2M
∂θd+2ψt
∂θd˙
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
∂θb=0,2ψt
∂θd=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
∂θb:˙
b=2M
∂θJ˙
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=θ2ψt
∂θ2=c, (38)
He=θ2M
∂θJ˙
J=1
ρo3αtθ(1d)2U
J2˙
J, (39)
Hin =θ2M
∂θd˙
d=1
ρo3αtθU
J˙
d, (40)
We finally insert all terms in (27) and (31) into (24):
ρ∂ψ
b:˙
bρDint +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 ρHeρHi n, (43)
To complete the formulation of a thermo-mechanical boundary value problem in a deformed body ,
we present the boundary conditions on its boundary Γ. We decompose the mechanical boundary into three
parts as Γ=ΓuΓtΓcwith ΓuΓt=ΓuΓc=ΓtΓc=, where Γuis the displacement boundary,
Γtis the traction boundary and Γcis the contact boundary. We decompose the thermal boundary into four
parts as Γ=ΓθΓqΓhΓcwith ΓθΓq=ΓθΓh=ΓqΓh=, 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(θθ)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 θis 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 :
Zw·ρad=Zxw:σd+Zw·ρbd+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
ifint
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
1yrn
2y
rn
2xrn
1x+ (Si(xp
2)Si(xp
4)rn
1y+rn
2y
rn
1xrn
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:
Zwρc˙
θd=Zxw·qdZwHed+Zw(Dint − Hin)d+
ZΓq
w¯
qdΓ+ZΓh
wh(θθ)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(θθ)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+κpt
i
˙
θn+1
ixSip , (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= (1ζ) 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 pth 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) = 13¯
r2+2¯
r3for 0 ¯
r1 and ω(¯
r) = 0 elsewhere;
¯
r=||xxp||/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
|xxi|<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
|ˆn1iˆn2i|(ˆn1iˆn2i),n2i:=n1iwith ˆnki =kp mpxSip
|kp mpxSi 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
iˆvki)·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
iˆvki)·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]k1
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
pcnSidΓ1
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 (θ2iθ1i:= [θ]1ik=1,
θ1iθ2i:= [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|,˙
λ0, ˙
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(δtδp
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(δtδp
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
Cipmpcpθn+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
1ivcm
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=1ν2
1
E1+1ν2
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 sb(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 =50C,θcool =20C. 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=5×108s,
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 ×105
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 = (e5d1)/(e51). As shown in Fig. 8, we successfully capture a crack propagating along the
direction with an angle around 67starting 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 ×108s.
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 =800C 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 =20C. 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 900C 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 ×105
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 108s. 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 ×105
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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