Applying FLD Damage Criteria to Predict Damage Evolution in

Polymer Sheets Under Nonlinear Forming Conditions

REZA SHAMIM

School of Aeronautics,

Northwestern Polytechnical University,

Xi’an, Shaan’xi 710072,

CHINA

ORCiD: https://orcid.org/0000-0002-4553-4968

Abstract: - Aerospace parts and other thin-walled constructions with intricate patterns frequently employ sheet

polymers. Formability is a crucial consideration since these materials may show damage and fractures during

the forming process. Predicting failure modes and comprehending formation limitations are paramount to

process design engineers. To forecast damage progression in single-sheet polymer forming processes that are

characterized by intricate and nonlinear strain patterns, this work uses a fully integrated elastic-plastic damage

model. For components with complex strain trajectories, the model accurately predicts deformation and damage

behavior, drawing on theories of finite strain and plane stress plastic deformation. When complicated and

nonlinear strain circumstances are present in sheet polymer forming procedures, the combination of finite

element analysis with continuum damage mechanics provides a quick and precise way to predict the damage

progression.

Key-Words: - Damage evolution, Forming limit diagram, Forming process, Nonlinear deformation, Polymer,

Simulation.

Received: April 23, 2024. Revised: October 27, 2024. Accepted: November 14, 2024. Published: December 31, 2024.

1 Introduction

One of the biggest challenges in material forming

processes is the prediction and analysis of rupture. It

is currently unclear how to accurately estimate when

a macrocrack will begin. Defect prediction can now

be completed faster than before, mostly due to

numerical software. Continuum damage mechanics,

which is a local approach to failure, is one of the

potential solutions this problem might be solved

with [1], [2], [3]. One measure of the material's

effective internal degradation after it has been

subjected to a local loading is called the damage

variable in a represented volume element (RVE).

Fracture in an RVE is defined after the last stage of

damage progression is identified by a local criterion,

[4], [5]. There were some voids and fissures in the

materials' microstructure. Most of the time, small

flaws grew when the loads reached a particular

point. Damage is the build-up of micro stresses

around faults or interfaces, which causes bonds to

break at the microscale level. The formation and

aggregation of microcracks or microvoids that

collectively start a single crack at the mesoscale

level is damage. At last, this represents the crack's

expansion on a macroscale. Studying the

development of faults and how they affect a

material's mechanical strength is the primary

objective of damage mechanics, [6]. We should take

into account the elastic-plastic and damage models

in the constitutive equations since significant plastic

deformation is accompanied by internal

deterioration of bonds (damage). In this study,

Lemaitre's model for ductile damage progression is

integrated with the isotropic elastic-plastic material

model. The derived constitutive equations are then

applied to predict the initiation and development of

fractures during sheet polymer forming processes.

The application of the Forming Limit Diagram

(FLD) damage criteria to predict damage evolution

in polymer sheets under nonlinear forming

conditions is a complex task that necessitates a

thorough understanding of damage mechanics and

material behavior. The FLD is traditionally used to

assess the formability of sheet metals, but its

application to polymers, particularly under nonlinear

conditions, requires careful consideration of the

unique damage mechanisms that occur in these

materials, [7], [8], [9], [10]. Damage evolution in

materials, including polymers, typically follows a

sequence of stages: nucleation, growth, and

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

161

Volume 19, 2024

coalescence of microvoids. The Gurson-Tvergaard-

Needleman (GTN) damage model is particularly

effective in describing these processes, as it

accounts for the void volume fraction, which is

critical in predicting mechanical properties and

potential defects such as rupture and wrinkling

during forming processes, [11], [12]. The GTN

model has been successfully applied to various

materials, including high-strength steels and

polymers, demonstrating its versatility in predicting

damage under different loading conditions, [13],

[14]. In the context of polymer composites, the

temperature and damage-dependent tensile strength

play a significant role in their formability. The

TDDTS model, which incorporates the effects of

temperature and damage, highlights the importance

of understanding how these factors influence the

mechanical behavior of fiber/polymer composites,

[15], [16]. This is particularly relevant when

considering the nonlinear forming conditions that

can lead to complex damage patterns not captured

by traditional FLD approaches. Moreover,

continuous damage mechanics provides a robust

framework for modeling the degradation of

polymers. This approach allows for the

transformation of geometric discontinuities into the

evolution of macroscopic mechanical properties,

which is essential for accurately predicting damage

under nonlinear conditions, [17], [18]. The

integration of Continuum Damage Mechanics with

the GTN model can enhance the predictive

capabilities for damage evolution in polymer sheets,

especially when subjected to varying strain paths

and loading conditions. The limitations of

conventional FLD models in estimating direct

fracture and deformation histories under nonlinear

strain paths have been noted in recent studies. For

instance, the need for improvements in necking-

based failure criteria has been emphasized, as these

models often fail to account for the complex

interactions that occur in advanced materials like

polymers, [19], [20]. The hybrid damage prediction

procedure, utilizing a stiffness degradation model

and an energy approach, accurately predicts the

damage of composite laminates under spectra

loading, showing excellent agreement with

experimental results, [21].

The incorporation of anisotropic damage

models, such as those modified to account for

material-induced anisotropic damage, can further

refine predictions of formability and damage

evolution in polymer sheets, [22], [23]. These

findings contribute to human construction and

modern environmental Science by enhancing the

durability and resilience of polymer-based

components used in sustainable infrastructure and

eco-friendly transportation solutions, ultimately

reducing material waste and improving lifecycle

performance in environmentally sensitive

applications.

2 Ductile Damage Model

To begin, the principles of Continuum Damage

Mechanics are examined in the context of uniaxial

stress. In this scenario, isotropic damage is

considered to be uniformly distributed across the

Representative Volume Element (RVE), [24]. The

RVE's cross-sectional area is represented by A, with

the assumption that damage within the RVE consists

of both voids and cracks. The total area occupied by

these defects is denoted as , which leads to an

effective cross-sectional area for the RVE, labeled

as 

󰆻:

A A A

(1)

The definition of the damage variable  is then

introduced as:

AAA

DAA





(2)

First proposed by Kachanov, is the damage

variable. D is defined as 0 in an undamaged state

and 1 at full failure. Consequently, the expression is

given as:

01D

(3)

The RVE, in both its damaged and effective

undamaged configurations, is subjected to the same

tensile force . As a result, the effective uniaxial

stress can be determined using the following

expression:





(4)

where  is the true stress in the RVE. By

substituting the eq. 2 to eq. 4, the following equation

for the effective uniaxial stress is achieved:

 represents the true stress within the RVE. By

substituting equation (2) into equation (4), the

expression for the effective uniaxial stress is

obtained as follows:





(5)

In the context of irreversible thermodynamics,

the energy damage criterion is derived from the

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

162

Volume 19, 2024

damaged elasticity potential, which is part of the

state potential and kinetic law. Additionally, damage

evolution is determined from the dissipation

potential. Assuming small strains and displacements

during the thermodynamic process, damage is

treated as a state variable, [25], [26]. Consequently,

the Helmholtz free energy, , is regarded as a scalar

function of these state variables.

 

, , , , ,

T r D

    



(6)

Here, , and  denote the elastic and plastic

strain tensors related to the stress tensor, while 

represents the temperature connected to entropy

density. The variable  signifies the damage-

accumulated plastic strain associated with isotropic

strain hardening, and  is the back strain tensor

linked to kinematic hardening.  is related to the

defects of the sample. Assuming the density 

which serves as an approximation for ductile

damage, remains constant, and applying the second

law of thermodynamics through the Clausius-

Duhem inequality, the relevant variables for an

isothermal process can be derived as follows:

 

F A A











(7)











In the case of yielding considering the

equivalent strain statement the  function will be

represented as (8) equations, where  is the strain-

hardening variable, and  is the back stress tensor.

1D v y

F X R



    



(8)

 

y v D

R X D



   

(9)

Here,  is referred to as the strain energy

release rate resulting from the stiffness reduction in

the RVE where damage has occurred. This variable

is also the key factor driving the damage

phenomenon.

p pD D



  

(10)

DD





and





are the equivalent plastic strains at the

onset of soft damage for the same biaxial tension

and compression tests. For isotropic materials, the

three-axis stress ratio is the same in the two-axis

tension state and the same in the two-axis

compression state.









and

are a function of

the strain rate material.

 

   

 

sinh sinh

sinh

   



   

  

   







     

 

(11)

In this expression,  denotes the Young's

modulus, and  represents the Von Mises

equivalent stress.  is a function of the triaxiality

ratio and in this context,  represents Poisson's ratio

and

H





is the triaxiality ratio, where 

denotes the hydrostatic stress. Based on Lemaitre's

damage criterion, the damage evolution equations in

terms of internal variables are:

 

01

eq eq





(12)

D is a damage variable that increases uniformly

with plastic deformation and is calculated in the

following way during the analysis of its positive

changes at each stage.

Δ0



(13)

The change initiation criterion continues until the

following condition is satisfied.

major





(14)

 

DrE D





(15)

Here, 󰇗 refers to the plastic consistency parameter,

while  is the material-specific damage parameter.

3 Numerical Simulations

To solve nonlinear problems, as a rule, numerical

methods are used, which include the finite element

method. Currently, FEM is the most popular way to

solve practical problems of the mechanics of a

deformable solid, [27]. With its help, calculations

are carried out to determine the stress-strain state

and the bearing capacity of real structures of various

industries and construction. In this case, problems of

both general and local strength are effectively

solved. The development of the finite element

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

163

Volume 19, 2024

method for dynamic and nonlinear problems

provides the ability to reliably model complex

processes such as fracture, impact, loss of stability,

stamping, drawing, etc. [28], [29], [30], [31], [32].

Almost all the problems of the solid-state solid-

propellant rocket engine were formulated and solved

in the framework of finite element techniques. To

highlight the core features of Lemaitre’s damage

model, a numerical example is presented involving a

rectangular deep drawing process. This simulation is

critical for validating the model due to its complex

strain paths, [33], [34], [35]. The verification was

carried out using an explicit finite element code,

simulating the complex mechanics of the deep

drawing process. In this process, a rectangular blank

undergoes significant plastic deformation to produce

a three-dimensional rectangular box shape. This

setup involves a deformable sheet, a rigid punch that

drives the forming process, a die that shapes the

material, and a blank holder that controls the

material flow. The simulation models these

components in 3D, capturing the interactions and

stresses across each element during deformation. To

optimize computational efficiency, the analysis

considers only one-quarter of the geometry,

leveraging symmetric boundary conditions. This

symmetry reduces computational load while

retaining full accuracy in the deformation behavior.

The die, characterized by a flat surface, features a

central rectangular opening measuring 120 mm by

190 mm, with rounded corners (radius of 45 mm) to

minimize stress concentration and rounded edges

(radius of 20 mm) to facilitate smoother material

flow. The rigid punch, square in shape and

dimensioned at 100 mm by 150 mm, mirrors the

rounding features of the die, ensuring a uniform

pressing force during the deep drawing operation.

The blank holder, also modeled as a rigid

component, is a flat surface that restricts the sheet’s

movement and applies controlled force along the

blank’s edges, limiting material flow to avoid

wrinkling and ensuring an even draw-in toward the

die cavity. Initially, the deformable blank is a flat

rectangular sheet with a thickness of 3 mm, which

undergoes deformation under the punch’s force,

forming the desired box shape within the specified

die cavity. The detailed schematic of this setup is

illustrated in Figure 1.

The blank interacts with the punch, die, and

blank holder, all modeled using a 10-node quadratic

tetrahedron (C3D10) element in Abaqus. This

element is ideal for accurately simulating complex

deformations in solid structures, utilizing quadratic

interpolation functions to capture intricate shapes

and linear deformations in structural analyses. The

distance between the blank holder and the die

surface is fixed at 0.5 mm. Friction coefficients are

0.18 for the interaction between the punch and the

blank and 0.10 for the die-holder and the blank.

ABS is the material properties applied for the blank,

and its damage characterizations are specified in

Table 1.

Fig. 1: The schematic of the problem

Table 1. Mechanical properties of ABS Plus

Ultimate strength (MPa)

Yield strength (MPa)

Elongation at yield (%)

Modulus of elasticity (GPa)

2.2

Poisson’s ratio

0.36

Density (kg/)

1020

Damage parameter, 

Damage parameter,  (MPa)

2.532

Critical damage parameter, 

0.434

A subroutine called user material (UMAT) was

created as a Fortran program to characterize the

material’s crystal plasticity behavior. ABAQUS, a

non-linear finite element program, was connected to

this function. Considering the material’s initial

texture and its progression over time, our approach

allowed us to model an arbitrary plastic

deformation. To satisfy the damage initiation

requirement, the following conditions must be met:

the material must experience sufficient levels of

stress and strain that exceed defined thresholds,

accounting for factors such as stress triaxiality and

strain rate. These parameters collectively indicate

when the material’s capacity to withstand further

deformation without fracturing has been surpassed,

triggering the onset of damage.

 

Δ0



Dpl pl



  

(16)

Collecting the FLD damage initiation criterion,

the ductile criterion is applied to predict the onset of

localized necking and eventual fracture in the

polymer sheet during the deep drawing process,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

164

Volume 19, 2024

capturing the material’s capacity to undergo plastic

deformation before failure.

 



major

FLD FLD

major minor f





(17)

In the forming limit diagram, the condition

   provides the damage initiation criterion.

The variable  is a function of the current

deformation state and is defined as the ratio of the

major limit strain on the forming limit diagram

evaluated at the current values of the minor

principal strain, , and predefined field

variables, 

, to the current major principal strain,

.

The below term provides the equivalent plastic

strain at damage initiation as a tabular function

between minor and major strains, optionally the

equivalent plastic strain rate, and predefined field

variables for the definition of the considered

damage initiation criterion.

 

major minor i

  

(18)

In the numerical simulation, the ratio of primary

strain rates is denoted by α. The software program

involved calculates the value of t across a specified

time increment, from t to ∆t, for each abrupt change

in the deformation path.

 

Δ1 Δ

    minor

major



    

(19)

The constitutive equations of rate-dependent

crystal plasticity for the slip magnitudes on 12 slip

systems are solved in this subroutine. Semi-implicit

time integration was used in the following way to

achieve this:

2t t dt

  

  









(20)

4 Results

The following images display the full simulations of

the deep drawing process for the drawing ratios

indicated above together with the blank's damage

distribution.

When the depth is 110 mm, the maximum

damage seen in Figure 2 is 0.998. This indicates that

the material is approaching its critical failure point.

As the depth increases, the localized stress

concentration intensifies, leading to progressive

material degradation. Beyond this depth, even slight

increases in loading could result in complete failure

or fracture, as the damage variable nears its upper

threshold. This observation highlights the

importance of considering material behavior at

critical depths in order to prevent catastrophic

failure under such conditions. The following images

display the full simulations of the deep drawing

process for the drawing ratios indicated above

together with the blank's damage distribution.

Fig. 2: Initial observation of damage occurrence in

the part under loading conditions

Figure 3 demonstrates that the point at which

the punch's corner produces a significant amount of

plastic deformation is where the rupture starts. It

shows the damage that occurred on the surface of

the designed model under an applied load of 69 kN

and a punch displacement of 115 mm. At this

loading condition, surface fractures and localized

material failure are evident, particularly in regions

with high-stress concentrations. The image clearly

illustrates the spread of damage across the surface,

with notable deformation near the contact points

between the punch and the model. These results

suggest that, as the displacement increases, the

material experiences significant strain, further

contributing to the onset of cracks and potential

failure zones within the structure.

Fig. 3: Identification of the specific damage location

within the part

In the vicinity of the blank's edge, wrinkling and

damage were also produced when the blank holder

force was uncontrolled. Figure 4 shows how the

material extends unevenly by illustrating the

connection between major and minor strain during

deformation. This diagram aids in locating stress

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

165

Volume 19, 2024

channels and forecasts failure modes in the

structure, such as necking and localized thinning.

The components of the ductile criterion function

as a measure of damage initiation, incorporating the

effects of stress triaxiality and strain rate to assess

the material's capacity to endure plastic deformation

before fracture where the equivalent plastic strain

rate is involved.

 

pl pl

  

(21)

where  represents the stress triaxiality, defined as

the ratio of pressure stress to Mises equivalent

stress.

/pq



(22)

Fig. 4: Relationship between major strain and minor

strain during deformation

Figure 5 depicts the equivalent plastic strain

path for each critical element of the blank in the

forming process, highlighting the elastic and

nonlinearity and the plastic strain progression over

the sample.

Fig. 5: Force-displacement curve illustrating the

mechanical response of the part under loading

Figure 4 and Figure 5 demonstrate a noticeable

discrepancy between the numerical simulations and

analytical solution results at the peak points for the

three examples (3, 5, and 7 mm), with percentage

differences of 0.79%, -1.58%, and 1.22%,

respectively.

Figure 6 displays two views of damage

distribution within the model. On the left, the

damage distribution at a depth of 123.5 mm is

shown, illustrating how damage progresses within

the material at this specific depth. The right side of

the figure presents the damage distribution at the

end of the process, with a final depth of 135 mm.

This comparison reveals the evolution of damage as

the process advances, highlighting the increase in

damage intensity and its spread throughout the

material. The visual contrast between the two depths

provides insight into the depth-dependent behavior

of damage and its impact on the material’s integrity

throughout the process.

Fig. 6: Left: Damage distribution in depth of 123.5

mm. Right: Damage distribution at the end of

process 135mm

5 Conclusion

This study integrates a fully connected elastic-

plastic-damage model with an explicit finite element

method, employing plane stress and finite strain

calculations to simulate the rectangular deep

drawing process. The analysis shows that the blank

holder's positioning effectively prevents wrinkling,

ensuring material integrity. At a depth of 110 mm,

maximum damage reached 0.998, indicating a near-

critical failure state, where minor loading increases

could lead to severe failures. Quantitative data

revealed that, under a punch load of 69 kN and a

displacement of 115 mm, surface fractures occurred

in high-stress regions, highlighting the model's

reliability in predicting damage initiation and

progression. Figures illustrate damage evolution,

showing localized damage at 123.5 mm compared to

a more widespread distribution at 135 mm,

underscoring the importance of depth-dependent

damage behavior. The model's predictive accuracy

is vital for optimizing forming processes in the

aerospace and automotive industries, enabling

engineers to adjust strain limits and deformation

paths to reduce fracture risks and extend the service

life of thin-walled polymer components. This

research advances damage prediction

methodologies, promoting more efficient and

sustainable manufacturing practices. Additionally,

numerical simulation combined with damage

continuum mechanics offers a quick and accurate

method for predicting damage evolution, rupture,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

166

Volume 19, 2024

and forming limit curve ratios in polymer forming

processes characterized by nonlinear and complex

strain paths. Parameters such as strain limits,

deformation paths, material properties, and stress

concentrations are essential for predicting the FLD

Damage Criteria and understanding damage

evolution in polymer sheets. Specifically, strain

limits define the threshold for material failure, while

deformation paths dictate the material's response to

applied loads. The inherent material properties,

including elasticity, plasticity, and damage

tolerance, influence the behavior of polymer sheets

under various loading conditions, and stress

concentrations at geometric discontinuities

significantly impact damage initiation and

progression.

Declaration of Generative AI and AI-assisted

Technologies in the Writing Process

During the preparation of this work the author used

Grammarly for grammar and spell-checking and

ChatGPT to improve readability and refine wording.

After using this tool/service, the author reviewed

and edited the content as needed and takes full

responsibility for the content of the publication.

References:

[1] Lemaitre, J., A Course on Damage

Mechanics, Springer Verlag, Berlin, 1992.

[2] Krajcinovic, D., Damage mechanics,

Mechanics of materials, Vol. 8, No. 2-3,

1989, pp. 117-197. DOI: 10.1016/0167-

6636(89)90011-2.

[3] Vala, J., On a computational smeared damage

approach to the analysis of strength of quasi-

brittle materials, WSEAS Transactions on

Applied and Theoretical Mechanics, Vol. 16,

2021, pp.283-292.

https://doi.org/10.37394/232011.2021.16.31.

[4] Irwin, G.R., Wells, AA., A Continuum-

Mechanics View of Crack Propagation,

Metallurgical Reviews, Vol. 10, No.1, 1997,

pp. 223-270. DOI:

10.1179/mtlr.1965.10.1.223.

[5] Kachanov, L., Introduction to continuum

damage mechanics, Springer Science &

Business Media, Vol. 10, 2013.

[6] Achineethongkham, K. and Uthaisangsuk, V.,

Analysis of forming limit behaviour of high

strength steels under non-linear strain paths

using a micromechanics damage modeling,

International Journal of Mechanical Sciences,

Vol. 183, 2020, pp.105828. DOI:

10.1016/j.ijmecsci.2020.105828.

[7] Basak, S. and Panda, S.K., Use of uncoupled

ductile damage models for fracture forming

limit prediction during two-stage forming of

aluminum sheet material, Journal of

Manufacturing Processes, Vol. 97, 2023,

pp.185-199. DOI:

10.1016/j.jmapro.2023.04.042.

[8] Bharti, S., Gupta, A., Krishnaswamy, H.,

Panigrahi, S.K., and Lee, M.G., Evaluation of

uncoupled ductile damage models for fracture

prediction in incremental sheet metal forming,

CIRP Journal of Manufacturing Science and

Technology, Vol. 37, 2022, pp.499-517. DOI:

10.1016/j.cirpj.2022.02.023.

[9] Zhang, M., Mirza, H.A., Zhang, Y.C.,

Tabasum, M.N., Lang, H.L. and Zou, P.Z.,

Investigation into a comprehensive forming

limit curve considering multi-parameters

effects for semi-cured FMLs through uniform

pressure loading test, International Journal of

Damage Mechanics, Vol 32, No. 2, 2023,

pp.289-317. DOI:

10.1177/10567895221135054.

[10] Tekkaya, A.E., Bouchard, P.O., Bruschi, S.

and Tasan, C.C., Damage in metal forming,

CIRP Annals, Vol. 69, No. 2, 2020, pp.600-

623. DOI: 10.1016/j.cirp.2020.05.005.

[11] Berezhnoi, D. V., Shamim, M. R., Numerical

investigation of clinch connection

manufacturing process, Procedia

Engineering, Vol. 206, 2017, pp, 1056-1062.

DOI: 10.1016/j.proeng.2017.10.594.

[12] Karim, B., Mohand, O. O., Nasereddine, Z.,

Sébastien, T., Investigation of the influence of

incremental sheet forming process parameters

using response surface methodology,

Metallurgical Research & Technology, Vol.

118, No. 4, 2021, pp 401. DOI:

10.1051/metal/2021039.

[13] Yoon, J. S., Noh, H. G., Song, W. J., Kang, B.

S., Kim, J., Comparison of experimental and

numerical failure criterion for formability

prediction of automotive part, Advanced

Materials Research, No. 658, 2013, pp. 354-

360. DOI:

10.4028/www.scientific.net/AMR.658.354.

[14] Isavand, S. and Assempour, A., Effects of

microstructural morphology on formability,

strain localization, and damage of ferrite-

pearlite steels: experimental and

micromechanical approaches, Metallurgical

and Materials Transactions A, Vol. 52, 2021,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

167

Volume 19, 2024

pp.711-725. DOI: 10.1007/s11661-020-

06115-2.

[15] Nasir, M. W., Chalal, H., Abed-Meraim, F.,

Formability prediction using bifurcation

criteria and GTN damage model,

International Journal of Mechanical Sciences,

No. 191, 2021, pp. 106083. DOI:

10.1016/j.ijmecsci.2020.106083.

[16] Li, Y., Li, W., Zhang, X., Shao, J., Deng, Y.,

Modeling and influencing factor analysis for

the temperature and damage dependent tensile

strength of fiber/polymer composites,

Polymer Composites, Vol. 43, No. 8, 2022,

pp. 5573-5581. DOI: 10.1002/pc.26869.

[17] Zirakashvili, N., On the numerical solution of

some two-dimensional non-classical elasticity

problems by boundary element method,

WSEAS Transactions on Applied and

Theoretical Mechanics, Vol. 14, 2019, pp.17-

27.

[18] Su, J., Cheng, J., Zhao, G., Hu, X., Tao, J.,

Degradation simulation of Poly (L-lactic acid)

(PLLA) vascular stents based on the

continuous damage model, Journal of

Physics: Conference Series, Vol. 2256, No. 1,

2022. pp. 012027. DOI: 10.1088/1742-

6596/2256/1/012027.

[19] Karyakin, M. and Obrezkov, L., Radial

Compression Stability of a Nonlinearly

Elastic Cylinder with Internal Stresses,

WSEAS Transactions on Applied and

Theoretical Mechanics, Vol. 14, 2019,

pp.159-163.

[20] Pereira, R., Peixinho, N., Sérgio L. Costa. S.

L., A Review of Sheet Metal Forming

Evaluation of Advanced High-Strength Steels

(AHSS), Metals, Vol. 14, No. 4, 2024, pp 394.

DOI: 10.3390/met14040394.

[21] Khavin, G., Zhiwen, H., Predictions of tool

wear by estimating weight loss during

polymer composites processing, Design,

construction, maintenance, Vol. 3, 2023, pp.

293–300. DOI: 10.37394/232022.2023.3.29.

[22] Bousfia, M., Aboussaleh, M., Ouhbi, B.,

Hybrid damage prediction procedure for

composite laminates submitted to spectra

loading, WSEAS transactions on applied and

theoretical mechanics, Vol. 15, 2021, pp.

248–257.

https://doi.org/10.37394/232011.2020.15.27.

[23] Niazi, M. S., Wisselink, H. H. Meinders, V.

T., Van Den Boogaard, A. H., Material-

induced anisotropic damage in DP600,

International Journal of damage mechanics,

Vol. 22, No. 7, 2013, pp. 1039-1070. DOI:

10.1177/1056789512468914.

[24] Egorova, O.V. and Hein, T.Z., Improvement

of mechanical properties of polymer materials

by the nanosized ceramic particles, WSEAS

Transactions on Applied and Theoretical

Mechanics, Vol. 16, 2021, pp.134-141.

https://doi.org/10.37394/232011.2021.16.14.

[25] Öchsner, A., Continuum damage mechanics,

Springer Singapore, 2016.

[26] Rynkovskaya, M., Alexandrov, S. and

Elberdov, T., Plane Strain Expansion of a

Cylindrical Cavity in an Infinite Porous

Rigid/Plastic Medium Obeying a General

Yield Criterion, WSEAS Transactions on

Applied and Theoretical Mechanics, Vol. 18,

2023, pp.318-326.

https://doi.org/10.37394/232011.2023.18.30.

[27] Shamim, R., Exploring the impact on contact

adhesion layer properties in numerical

simulations, PLoS ONE, Vol. 19, No.10,

2024. DOI: 10.1371/journal.pone.0312436.

[28] Guo, Y.M. and Moriki, K., The Elastic-Plastic

Micromechanical Response of the Particle

Volume Fraction and the Average Size of

Particles, WSEAS Transactions on Applied

and Theoretical Mechanics, Vol. 14, 2019,

pp.82-89.

[29] Shahzamanian, M.M. and Wu, P.D., Study of

forming limit diagram (FLD) prediction of

anisotropic sheet metals using Gurson model

in MK method, International Journal of

Material Forming, Vol. 14, 2021, pp.1031-

1041. DOI: 10.1007/s12289-021-01619-7.

[30] Vobejda, R., Šebek, F., Kubík, P. and

Petruška, J., Solution to problems caused by

associated non-quadratic yield functions with

respect to the ductile fracture, International

Journal of Plasticity, Vol. 154, 2022,

pp.103301. DOI:

10.1016/j.ijplas.2022.103301.

[31] Lee, H.W. and Basaran, C., A review of

damage, void evolution, and fatigue life

prediction models, Metals, Vol. 11, No. 4,

2021, pp.609. DOI: 10.3390/met11040609.

[32] Shamim, R., Investigation of bending and

crush behaviors in polymer lattice structures:

Computational approaches and experimental

evaluation, Journal of Reinforced Plastics and

Composites, 2024, pp. 07316844241273043.

DOI: 10.1177/07316844241273043.

[33] Fincato, R. and Tsutsumi, S., Coupled elasto-

viscoplastic and damage model accounting for

plastic anisotropy and damage evolution

dependent on loading conditions, Computer

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

168

Volume 19, 2024

Methods in Applied Mechanics and

Engineering, Vol. 387, 2021, pp.114165.

DOI: 10.1016/j.cma.2021.114165.

[34] Rong, H., Hu, P., Ying, L., Hou, W. and Dai,

M., Modeling the anisotropic plasticity and

damage of AA7075 alloy in hot forming,

International Journal of Mechanical Sciences,

Vol. 215, 2022, pp.106951. DOI:

10.1016/j.ijmecsci.2021.106951.

[35] Voyiadjis, G. Z., Continuum damage

mechanics, Materials Modeling: Methods,

Dordrecht: Springer Netherlands, 2005, pp.

1183-1192.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Reza Shamim: Investigation; methodology; data

curation; writing original draft; conceptualization;

formal analysis, resources.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The author received no financial support for the

research, authorship, and/or publication of this

article.

Conflict of Interest

The authors declare that they have no known

competing financial interests or personal

relationships that could have appeared to influence

the work reported in this paper.

Data availability statement

The data that support the findings of this study are

available from the corresponding author upon

request.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2024.19.18

Reza Shamim

E-ISSN: 2224-3429

169

Volume 19, 2024