Transactions Template


 

  

JOURNAL OF ENGINEERING RESEARCH AND TECHNOLOGY, VOLUME 8, ISSUE 1, MARCH  2021 

 

  

1  

A Stress-State based Peridynamics model for elastio-plastic 

material modeling 

Mahmoud M. Jahjouh – University College of Applied Sciences UCAS 
https://doi.org/10.33976/JERT.8.1/2021/1   

 
Abstract—A Stress-State based PD (SSPD) model using a well-known yield criteria is proposed in this paper 
and tested on the modeling of two dimensional bars under different loading levels as a first step for further 
development. SSPD is based upon peridynamics (PD) which utilize temporal spatial integro-differential equation 
of motion and formulates continuum problems in terms of integral equations, which are capable of modeling 
discontinuities such as cracks. The proposed bond strength not only depends on the bond stretch, but on the 
current state of all bonds connected to a particle as well. Thus, a stress-based peridynamics model is obtained. 
The tensile simulation compared to conventional FEM shows promising performance with an error of 5%. 
Compression simulations, however, need more investigation to include the effect of contact forces. 

Index Terms—Peridynamics, Stress-State, Modeling, FEM.  

 

I INTRODUCTION

 
Peridynamics (PD) is a recently-developed novel contin-

uum mechanics that strives to unify the distinction between 

"discrete" and "continuous" media currently present in the 

classical continuum mechanics theory [1] 

 

Since its development in the landmark paper [1], PD was 

used to model problems where classical continuum mechan-

ics, which rely on partial differential equations, fail to do so. 

The majority of research in classical continuum mechanics 

tries to amend the theory by "adding-on" terms or equations, 

which usually results in a modified theory with limited ap-

plication for a certain class of problems [2]. 

 

One of the most prominent of such additions is eXtended 

Finite Element Method (XFEM) presented in Moes et al. [3] 

and Fries and Belytschko [4] as an extension to conventional 

Finite Element Method (FEM) to capture discontinuities. 

PD, on the other hand, inherently provides a suitable frame-

work that predicts material failure and discontinuities. A 

comparison between PD and XFEM can be found in the 

work of Agwai et al. [5]. 

 

Nevertheless, PD has been successfully implemented in 

various problems investigating a continuum with discontinu-

ities.  

 

The first bond-based PD model was presented in the 

landmark paper by Silling [1] and further explained in [6]. 

Yet, bond-based PD model suffer from the limitations im-

posed on incorporating the Poisson's ratio. The introduction 

of a state-based PD model by Silling et al. [7] and Silling 

and Lehoucq [8] presented a method of dealing with such 

limitation [9,10]. 

 

PD was also successfully applied in both material and 

geometric non-linear analysis. The behavior of elastic, plas-

tic and damaged models based on PD was investigated by 

Gerstle [11] 

 

Most recently, an ordinary state-based peridynamic mod-

el was proposed for the analysis of models with geometrical 

non-linearities [12]. The bond stretch was defined using 

logarithmic functions suitable for large deformations. 

 

The flexibility and simplicity of PD enabled it to be cou-

ploed within a finite element analysis. The work of Kilic and 

Madenci [13] used PD to model the regions of discontinui-

ties within the displacement field and coupled those with 

FEM in an attempt to take advantage of both methods. Simi-

lar research can be found in [14, 15, 16]. PD was also im-

plemented in a FEA framework in [17, 18, 19, 20]. 

 

 

Interested readers may consult extensive literature sur-

veys provided by Madenci and Oterkus [21] and Javili [22]. 

 

PD is, however, still not a flawless theory. As will be 

shown in section (II), a continuum is modeled via discrete 

particles connected via bonds that can be modeled as 

springs. The derivation of the spring constant relying on the 

bulk modulus seems to be heuristic [21].  

 

Engineers are used to using the Young's Modulus, also 

known as the Elastic Modulus, in describing tensile tests and 

its resulting stress-strain relationships. The research present-

ed in this contribution derives bond forces within a PD state 

based on a well-established yield criteria, which enables a 

https://doi.org/10.33976/JERT.8.1/2021/1


Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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deeper understanding of how bonds "perform" and simpli-

fies the implementation of advanced analysis via PD such as 

fatigue. 

 

The research presented in this paper continues with a 

brief introduction of the general PD approach in section (II), 

followed by the derivation of the governing equations for 

PD using different yield criteria in section (III). The numeri-

cal procedure of analysis is presented in section (IV) fol-

lowed by the case studies in section (V). Finally, conclusions 

and recommendations are drawn in section (VI). 

II BOND-BASED PERIDYNAMICS 

Peridynamics relies on a temporal spatial integro-

differential equation of motion [1]. Thus, in a PD model, the 

body being modeled is discritized into a set of particles with 

differential volumes. The response of a body 𝓡, shown in 
Figure 1, to external forces is assumed to depend on the 

displacement of particles relative to their initial positions in 

the reference configuration [1]. 

 
The internal forces affecting a particle in PD are derived 

from a number of particles in its vicinity [22]. Thus, the 

particle cannot interact with other particles beyond a horizon 

𝛿  i.e. a particle 𝐱 in the reference configuration can only 
interact with another particle 𝐱′ that lies within a neighbor-
hood 𝓗𝐱 of 𝐱 defined as 

 

𝓗𝐱 = {𝐱 ∈ 𝓡 ∶|𝐱
′ − 𝐱| < δ → 𝐱′ ∈ 𝓡}      (1) 

 
The basis of PD is the integral equation 𝐋𝐮(𝐱, 𝑡) which re-
lates to the force per unit volume 𝐟 of a particle 𝐱 at time 𝑡 
resulting from the interaction of all other particles 𝐱′ ∈ 𝓗𝐱 
[1]. This integral equation is defined as 

 

𝐋𝐮(𝐱, 𝑡) =  ∫ 𝑓(𝜂, 𝜉)𝑑𝑉𝜂      ∀
𝓗𝐱

𝐱 ∈ 𝓗𝐱′    (2) 

 
As indicated in eq. (2), a bond force 𝑓 exists between two 
particles 𝐱′ and 𝐱 that is defined by the relative position 
𝝃 =  𝐱′ − 𝐱  and the relative displacement 𝝃 =  𝐮(𝐱′, 𝑡) −

𝒖(𝐱, t)  where 𝐮 is the displacement field at time 𝑡. 
 
The PD equation of motion [1] is defined as 

 

𝝆(𝐱)�̈�(𝐱, 𝑡) = 𝐋𝐮(𝐱, 𝑡) − 𝒃(𝐱, 𝑡)    (3) 
 
where 𝒃(𝐱, 𝑡) represents the body forces on particle 𝐱 at 
time 𝑡. 
 

One of the assumptions that can be used when model-
ing the bond force 𝑓 in PD is - in its simplest form - a 
linear spring. 

III STRESS-STATE BASED PD (SSPD) 

The PD theory formulates continuum problems in terms 

of integral equations unlike its classical counterparts, which 

relies heavily on partial differential equations, which - as 

previously mentioned - face difficulties whenever disconti-

nuities are present in the continuum. One of the most com-

mon discontinuities is crack propagation and growth. 

 

Cracks are considered in PD as bond breakage [2], which 

requires the definition of a limiting bond stretch that, when 

exceeded, results in a bond failure. 

 
However, the definition of PD bonds as linear - or even 

non-linear - springs represents a somehow heuristic ap-
proach. Thus, the proposed model in this research formu-
lates the bond forces using well-known stress-strain rela-
tionships and models the material behavior using estab-
lished yield criteria. 

 
The proposed bond strength depends not only on the 

bond stretch, called strain in this research, but on the 
current state of all bonds connected to the particle as well. 
Thus, a Stress-State based PD (SSPD) is realized. 

 
The bond strength between two particles in an SSPD, 𝑓 

can be calculated as 
 

𝑓(𝜂, 𝜉) = 𝐴𝑒
|𝜂| − |𝜉|

|𝜉|
𝐸

𝜂

|𝜂|
≤ 𝐴𝑒𝜖𝑦 𝐸

𝜂

|𝜂|
     (4) 

 
where 𝐸 is the modulus of elasticity (young's modu-

lus), 𝜖𝑦  is the strain at yield and 𝐴𝑒  is the effective area 
allocated to the bond. The bond described in eq. (4) is 
subjected to the strain 

|𝜂|−|𝜉|

|𝜉|
 not exceeding the rupture 

strain 𝜖𝑟   which is a characteristic value in a uni-axial 
tension test. If the strain between two particles exceeds 𝜖𝑟 , 
the bond is broken. Broken bonds in the presented work 
cannot be healed. 

 
The implication of considering strain and young's 

modulus in eq. (4) is that the behavior of materials sub-
jected to tensile stresses are to be incorporated in the 
SSPD. Such behavior is usually characterized by a linear 

 

Fig.1. Schematics of a body 𝓡. 
PD2-figure0.pdf

 

 



Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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part, followed by a non-linear part. A typical stress-strain 
behavior for steel is shown in Figure 2. To simplify such 

behavior, strain hardening is neglected and a bi-linear 
stress-strain curve shown in Figure 3 is adopted. 

 
The stresses are obtained from the bond force 𝑓 by di-

viding it by the "tributary area". In the research presented 
in this contribution, the area is assumed to be 

 
𝐴 = 𝑐 × Δ𝑥    (5) 

 
where Δ𝑥 is the lattice spacing and 𝑐 is the length of a 

side of a regular polygon inscribed in a circle of radius 𝑟𝑡𝑟  
and is calculated as 

𝑐 = 𝑟𝑡𝑟 √2 − 2 cos (
2𝜋

𝑛
)     (6) 

 
where 𝑛 is the number of bonds for the particle in 

question. 
 
A particle is usually bound to multiple particles in its 

vicinity, as shown in Figure 4. The bond forces described 
in eq. (4) can reach the forces obtained when stresses 
within the material reach yielding stresses 𝜎𝑦 . 

 

Having multiple bonds with stresses that could reach 
𝜎𝑦 leads to the conclusion that a particle could be subject-
ed to stresses that produce an equivalent stress that is 
well beyond 𝜎𝑦 , even if eq. (4) is limited to 𝜎𝑦 . 

 
 
The equivalent stress resulting from a two dimensional 

stress state can be calculated to be 
 

𝜎𝑣 = √𝜎1
2 − 𝜎1 𝜎2 + 𝜎2

2     (7) 

 
After calculating 𝜎𝑣  for a particle, two cases can 

emerge: 
Case 1: 𝜎𝑣  does not cause the material to yield. 
Case 2: 𝜎𝑣  causes the material to yield. 
 
The question whether 𝜎𝑣  causes the material to yield or 

not is based on the selected yielding criteria. For example, 
using the yielding criteria according to Tresca [23] would 
result in a yield-surface for two dimensional stresses as 
shown in Figure 5. 

 
If 𝜎𝑣  is outside the envelope defined by the yield crite-

rion, a return-mapping should be performed to correct 
𝜎𝑣 back to the yielding surface, as no stresses higher than 
the yield stress are acceptable, since strain hardening is 

not considered in the research presented here. Such re-
turn-mapping is shown in Figure 6. 
 

IV NUMERICAL IMPLEMENTATION 

The simulation of a material with SSPD starts by generat-

ing a reference configuration using a lattice. In the research 

presented here, an equally spaced lattice is adopted with a 

spacing defined to be Δ𝑥, resulting in particles with a vol-
ume of Δ𝑥 3. 

 

Fig. 2. Typical Stress Strain Curve for Steel. 
PD2-figure0.pdf

 

 

 

Fig. 6. Return-Mapping for invalid stress states.  
PD2-figure0.pdf

 

 

 

Fig. 3. Bi-Linear stress-strain curve.   
PD2-figure0.pdf

 

 

 

Fig. 5. Yielding Surface according to Tresca.   
PD2-figure0.pdf

 

 



Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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After defining the reference configuration, the material is 

defined by selecting 𝜖𝑦 , 𝜖𝑟 , 𝐸 and the yield criterion to be 
considered. 

 

As for integrating the equation of motion in eq. (3), the 

Verlet Integration scheme is used. 

 

The pseudo-code of SSPD is shown in algorithm (1), 

whereas the return mapping is shown in algorithm (2). 

V  CASE STUDIES 

Three study cases are presented to investigate the perfor-

mance of SSPD for 2D problems and gain first insights. Of 

interest are the force displacement behavior and failure pat-

terns. The former is important to check that the suggested 

stress-state peridynamics is able to capture the behavior of 

the test specimen and compare it with the well-known stress-

strain behavior obtained in practice, whereas the later is 

important to check whether the failure patterns do meet the 

expectations based on sound engineering judgment. 

 

For all the cases stated below, the horizon was set to be 𝛿 =
√2 Δ𝑥, i.e. for an equidistant lattice, a particle is bound to its 
neighbors from 8 directions. The radius required to calculate 

the "tributary area" of the bonds was taken to be 𝑟𝑡𝑟 =
0.583 Δ𝑥. 
 

Furthermore, the material of all cases is assumed to be steel, 

with 𝐸 = 200 GPa, 𝜖𝑦 = 0.002, 𝜖𝑟  = 0.025 and 𝜌 = 7850 
kg/m3. 

 

The analysis is performed under a gradually increasing ex-

ternal forces. This is realized using the function 

𝐹(𝑡) =
𝜎𝑡 (Δ𝑥)

2

1 + 𝑒
𝑎[−

20
𝑏

𝑡+10]
     (8) 

where 𝜎𝑡  is the targeted stress level. 
 

A typical stress history is shown in Figure 7. 

 

2D Bar subjected to Tension 

To start with, a simple bar, shown in Figure 8, is subjected 

to tensile forces and analyzed. 

 

The bar dimensions were 60 mm x 21 mm x 3mm and Δ𝑥 
was assumed to be 3 mm. The discretized bar is shown in 

Figure 9 in which the center of each element is shown as 

nodes and the bonds are shown as links. 

 

 

 

  

 
PD2-figure0.pdf

 

 

 

 

Fig. 9. Discretization of 2D Bar. 

 

Fig. 7. Typical external applied stress history. 

 

Figure 8. 2D Bar subjected to forces. 



Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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Case 1 - 𝝈𝒕 ≤  𝝈𝒚  
To test the accuracy of SSPD, a targetted stress level 𝜎𝑡  well 
below the yield stress  𝜎𝑦  is applied in eq. (8). Thus, 𝜎𝑡  = 
250 MPa is selected. 

 

The displacements resulting from the SSPD analysis of 

the aforementioned bar are shown in Figure 10 and Figure 

11 for the x-direction and y-direction respectively. For com-

parison purposes, the resulting displacements using a con-

ventional finite element analysis software (Autodesk Robot) 

is shown in Figure 12 and Figure 13, respectively. 

 

A qualitative comparison of the displacement fields 

shown in Figure 10 through 13 shows that both methods 

provide displacement fields that are similar.  

 

Furthermore, a quantitative comparison shows that the 

maximum displacement 𝑈𝑥  is 0.70mm and 0.73mm for 
SSPD and FEM respectively. This indicates an error of 3mm 

or 4.3%. 

 

Similarly, the maximum displacement 𝑈𝑦  is 0.042mm and 
0.040mm for SSPD and FEM, indicating an error of 

0.002mm or 5%. 

 

 

 

 

 

 

 

 

 

Case 2 - 𝝈𝒕 >  𝝈𝒚  
For the material characteristics defined in this research - 

𝐸 = 200 GPa, 𝜖𝑦  = 0.002 - the yield stress 𝜎𝑦  is expected to 
be 400 MPa. Thus, a stress level 𝜎𝑡  of 500 MPa is selected. 

 

Multiple strain gauges are defined along the length of the 

bar at the centerline of the bar, namely at 𝑥= [16.5, 31.5, 
46.5] mm. 

 

The resulting stress strain diagram measured at the strain 

gauges is shown in Figure 14. This stress-strain diagram was 

obtained just before failure, which is evident from the strain 

of the sensor at x = 46.5mm which approaches a value of 

0.025, which was the value set for 𝜖𝑟 .  
 

After failure, the stress strain diagram becomes chaotic, as 

the bar starts vibrating violently. The linear portion of the 

stress-strain diagram has a slope of 200 GPa, which corre-

sponds to the provided young's modulus. The inelastic part 

follows a profile corresponding to a force-driven tensile test 

with no strain hardening. 

 

The bar just after exceeding 𝜖𝑟  is shown in Figure 15. It can 
be seen that failure occurs near model non-discontinuities, 

i.e. near the boundary conditions and force application parti-

cles. 

 

 

 

Fig. 13. FEM Results for 𝑈𝑦 . 

 

Fig. 11.  FEM Results for 𝑈𝑥 . 

 

Fig. 10. SSPD Results for 𝑈𝑥. 

 

Fig. 12. SSPD Results for 𝑈𝑦 . 



Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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2D Bar subjected to Compression 

The same bar is used in a compression test. Here, a tar-

geted stress level 𝜖𝑡 of 500 MPa is assumed. Furthermore, 
identical SSPD parameters is used in this test. 

 

An identical stress strain and failure diagram as shown in 

Figure 14 and 15 is obtained. This, however, is not compati-

ble with usual stress strain diagrams obtained from the usual 

compression tests, which indicates an area for future investi-

gation. 

 

The SSPD fails to capture compression correctly mainly due 

to the absence of contact forces particles used in the PD 

analysis. In the current proposed SSPD, particles are al-

lowed to get "crushed", i.e. to have a distance of zero be-

tween its neighbors. Even more, the distance can actually 

become negative, which indicates that a particle has been 

crushed and even pierced through its neighbor. Such behav-

ior is non-physical and requires further investigation. A 

recommendation that can be given here is to include contact 

forces within the framework of SSPD [22]. This is going to 

be the subject of future investigations. 

 

 

2D Bar with notches subjected to Tension 

The bar described previously is notched from both sides and 

used in a tensile test. The bar is shown in Figure 16. The 

targeted stress level 𝜎𝑡  of 500 MPa is applied. 
 

 

The resulting stress strain diagram is shown in Figure 17, 

obtained just before failure. The linear portion of the stress-

strain diagram has a slope of 200 GPa, which corresponds to 

the provided young's modulus. The inelastic part follows a 

profile corresponding to a force-driven tensile test with no 

strain hardening. 

 

The bar just after exceeding 𝜖𝑟  is shown in Figure 18. It can 
be seen that failure occurs near model non-discontinuities, 

i.e. near the crack as well as boundary conditions and force 

application particles. 

 

 

 

 

 

 

Fig. 15. 2D Bar just after failure 

 

Fig. 18. 2D bar with notches after failure. 

 

Fig. 14. 2D Bar stress strain diagram. 

 

Fig. 16. Discretization of the 2D Bar with a notch. 

 

Fig. 17. stress strain diagram for bar with notches. 



Mahmoud M. Jahjouh / A Stress-State based Peridynamics model for elastio-plastic material modeling (2021)  

 

  

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CONCLUSION 

A newly developed stress-based peridynamics model 

(SSPD) was presented in the research work of this contribu-

tion and applied on three numerical cases: 

 The first numerical case applies stresses well below the 

yielding stresses and shows that the proposed SSPD 

conforms well with FEM results with an error of 

around $5\%$. Furthermore, it shows the application of 

stresses well beyond the yielding stresses, resulting in 

stress-strain diagrams that conform with the expected 

stress strain diagram when a bilinear stress-strain rela-

tion ship is assumed. 

 The second numerical case shows a bar under com-

pression. The results are identical to those obtained 

from tension stresses. 

 The SSPD fails to capture compression correctly main-

ly due to the absence of contact forces particles used in 

the PD analysis. A recommendation that can be given 

here is to include contact forces within the framework 

of SSPD. This is going to be the subject of future in-

vestigations. 

 The third numerical case show a bar with notches to 

encourage failure near those notches. The failure does 

indeed occur near those notches, but still occur near the 

force application points too. 

 

For future research, it is planned to investigate the inaccu-

racies with regard to compression simulation and derive 

ways of including contact forces to prevent the ``crushing'' 

of the mesh. 

 

Also, implementing a stress-strain diagram that features 

strain hardening, as well as cyclic load analysis and fatigue. 

 

Furthermore, tests with 3D bar models are planned to be 

implemented next and compared to the results obtained form 

2D bars. 

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Mahmoud M. Jahjouh is an Assistant Professor at the Universitz 

College of Applied Sciences (UCAS) and is the head of its Engi-

neering and Information Systems department. Dr. Jahjouh has 

obtained his Ph.D. in Structural Engineering from the Leibniz 

University of Hannover, and is interested in topics with regard to 

structural optimization, shape optimization, modeling and damage 

detection.