Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Antarvedi Goda, Bharath
Doctor of Philosophy - PhD
University of British Columbia
Cutting mechanics of soft solids has been of growing interest due to its promising benefits in material characterization and other industrial applications. However, a full understanding of the physical phenomena is still missing, and several questions remain outstanding. In this thesis, we explore the simple problem of wire cutting. Through finite element analysis and dimensional arguments, we obtain a simple scaling relation between the wire radius and the steady-state cutting forces. We note that, in the absence of dissipative mechanisms, steady-state cutting forces are independent of blade sharpness. However, in the presence of dissipation in the form of friction and adhesion, we show that tough materials are sensitive to friction, whereas, brittle materials are sensitive to dissipation in the form of adhesion and wear. In the regime dominated by friction, cutting forces have a weaker sensitivity to blade sharpness. But, in the regime dominated by adhesion and shear damage/ wear, we have a linear dependence between cutting force and blade sharpness. By accounting for strain-stiffening and compressibility of the soft material, we note that tough and highly strain-stiffening materials are most sensitive to frictional dissipation. Whereas, in-compressible brittle materials are most sensitive to adhesion and wear dissipation. Finally, this thesis provides important findings: Toughness can be estimated from wire-cutting experiments from the intercept of the forcewire radius linear correlation, however this only estimates cleavage toughness. In the adhesive/wear regime, a larger radius would scale the cutting fracture process zone giving rise to higher apparent toughness.
Cutting Mechanics of Soft SolidsbyBharath Antarvedi GodaM.S. in Mechanical Engineering, University of Colorado Boulder, 2019B.E. in Mechanical Engineering, Anna University, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)August 2024© Bharath Antarvedi Goda, 2024The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Cutting Mechanics of Soft Solidssubmitted by Bharath Antarvedi Goda in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mechanical Engineering.Examining Committee:Mattia Bacca, Associate Professor, Mechanical Engineering, UBCSupervisorXiaoliang Jin, Associate Professor, Mechanical Engineering, UBCSupervisory Committee MemberReza Vaziri, Professor, Civil Engineering, UBCUniversity ExaminerSavvas G. Hatzikiriakos, Professor, Chemical and Biological Engineering, UBCUniversity ExaminerAnand Jagota, Professor, Chemical and Biomolecular Engineering, Lehigh Uni-versityExternal ExaminerAdditional Supervisory Committee Members:Jasmin Jelovica, Associate Professor, Mechanical Engineering, UBCSupervisory Committee MemberSrikantha Anasavarapu, Professor, Mechanical Engineering, UBCSupervisory Committee MemberiiAbstractCutting mechanics of soft solids has been of growing interest due to its promisingbenefits in material characterization and other industrial applications. However, afull understanding of the physical phenomena is still missing, and several ques-tions remain outstanding. In this thesis, we explore the simple problem of wirecutting. Through finite element analysis and dimensional arguments, we obtain asimple scaling relation between the wire radius and the steady-state cutting forces.We note that, in the absence of dissipative mechanisms, steady-state cutting forcesare independent of blade sharpness. However, in the presence of dissipation in theform of friction and adhesion, we show that tough materials are sensitive to friction,whereas, brittle materials are sensitive to dissipation in the form of adhesion andwear. In the regime dominated by friction, cutting forces have a weaker sensitivityto blade sharpness. But, in the regime dominated by adhesion and shear dam-age/wear, we have a linear dependence between cutting force and blade sharpness.By accounting for strain-stiffening and compressibility of the soft material, we notethat tough and highly strain-stiffening materials are most sensitive to frictional dis-sipation. Whereas, in-compressible brittle materials are most sensitive to adhesionand wear dissipation. Finally, this thesis provides important findings: Toughnesscan be estimated from wire-cutting experiments from the intercept of the force-wire radius linear correlation, however this only estimates cleavage toughness. Inthe adhesive/wear regime, a larger radius would scale the cutting fracture processzone giving rise to higher apparent toughness.iiiLay SummaryCutting is ubiquitous in our everyday life and has wide-ranging applications. De-spite this importance, we still do not fully understand the physical mechanisms thatgovern the cut response and the forces. This thesis proposes a simple mechanicalmodel to predict the force required to cut into a soft material by a rigid cylindricalwire as a function of material properties, the cutting tool geometry, and contactdissipation due to friction, adhesion, and wear. Finally, this thesis includes experi-mental validation and discusses the role of friction and adhesion. This sheds lighton microscopic mechanisms that govern cutting in different materials.ivPrefaceThe research presented in the current thesis is conducted by the author, BharathAntarvedi Goda, under the supervision of Prof. Mattia Bacca. The findings inchapters 2 and 3 of this thesis are published in a peer-reviewed journal: BA Goda,Z Ma, S Fregonese, and M Bacca, Cutting of Soft Matter - Scaling relations con-trolled by toughness, friction, and wear, Soft Matter, 2024. Contents of chapter 4are undergoing revisions as journal manuscripts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Soft Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Fracture Mechanisms in Soft Materials . . . . . . . . . . 41.3 Strain-Stiffening . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 16vi2.1 Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . 162.1.1 K-field . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Fracto-cohesive and Elasto-cohesive Length Scales . . . . . . . . 202.3 Nonlinear Elastic Fracture Mechanics . . . . . . . . . . . . . . . 232.3.1 Pure Shear . . . . . . . . . . . . . . . . . . . . . . . . . 243 Cutting Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Role of friction, adhesion and wear . . . . . . . . . . . . . . . . . 353.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Cutting Experiment . . . . . . . . . . . . . . . . . . . . . 423.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Role of Compressibility in Cutting . . . . . . . . . . . . . . . . . . . 474.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.1.1 Material Characterization . . . . . . . . . . . . . . . . . . 73viiList of TablesTable 3.1 Power-law coefficient n of the fitting function in Eq. (3.13),in the high toughness regime (Γo ≫ µRw) obtained to the leastsquare feet with an r-square accuracy of r2 ≥ 0.98 . . . . . . . 39Table 4.1 Power-law coefficient n of the fitting function in Eq. (4.9),in the high toughness regime (Γo ≫ µRw) for α = 5, α = 9obtained to the least square feet with an r-square accuracy ofr2 ≥ 0.98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viiiList of FiguresFigure 1.1 Different Stages in Cutting (I) Indentation (II) Cut initiation(III) Steady-state . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Material Classification based on stiffness: Siff/Hard or Soft,and based on toughness: Brittle or Ductile [5] . . . . . . . . . 3Figure 1.3 Mesh-like structure of an elastomer at nano-scales with cross-linking points denoted by x and the effective distance betweenthe crosslinks l√n . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.4 Different dissipative mechanisms in Soft Solids with FractureProcess Zone (FPZ) highlighted around the crack tip in greyand the typical size of FPZ marked below (source: [5]) . . . . 6Figure 1.5 (a) Dimensionless Cauchy stress σ22/µ vs remote stretch λ2(b) Dimensionless strain energy density Wf /µ vs remote stretchλ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.6 Relationship between cutting and fracture . . . . . . . . . . . 10Figure 1.7 (a) Schematics of Wire Cutting (b) Force (F) vs IndentationDepth (D) response showing different stages in cutting . . . . 12Figure 1.8 (a) Schematics of Y-shaped cutting setup (b) Striation in the cutsurface in the far-field moderate or high tearing regime (source:[30]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 1.9 (a) Cutting hard and soft solids (in grey) by blade (in black)(source: [5]) (b) Physical picture capturing the location of cracktip in relation to the blade tip while propagating the cut (source:[5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14ixFigure 2.1 Square plate of Length (L), with a center crack of length (2c),such that L≫ c, subject to a remote load σ∞ in X2. Polar coor-dinates (r,φ ) around the crack tip (Right) . . . . . . . . . . . 17Figure 2.2 (a) Plot of Cauchy stress (σ22) in X2 as a function of distancefrom the crack tip (r) at φ = 0o (b) Log-log plot of σ22 vs rhighlighting the validity of K-field in the region close to thecrack tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 (a) Typical Stress vs Strain response of a Linear-Elastic ma-terial with work of fracture Wf (area under the curve) shadedin blue (b) Typical Stress vs Strain response of a Nonlinear-Elastic material (c) Fracture Energy and Work of fracture/rup-ture Ashby plot (source: [10]) (d) Showing crack-opening pro-file of soft material with a critical crack opening displacementδctod = 7mm (source: [5]) . . . . . . . . . . . . . . . . . . . 22Figure 2.4 (a) FPZ and K-field zone around crack tip in LEFM (b) FPZ,Nonlinear Elastic zone, and K-field zone around crack tip inNLEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.5 (a) Pure Shear sample of Length (L) and Height (H) with an as-pect ratio (L/H = 100) and crack (in red) of length (c = 0.4L)snippet showing the mesh around the Crack tip (b) Results ofa simulation of Neo-Hookean sample with a remote stretch inX2 of λ2 = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.6 (a) Dimensionless Stress field (σ22/µ) evaluated near the cracktip of an Ogden hyperelastic material (α = 2,5) in Pure Shearusing FE simulations (b) Stretch field (λ2) evaluated near thecrack tip of an Ogden hyperelastic material (α = 2,5) in PureShear using FE simulations. . . . . . . . . . . . . . . . . . . 28Figure 3.1 Schematics of the setup adopted to perform wire cutting alongX1 on the soft hyperelastic sample (in grey) by a cylindricalwire (in brown) of radius Rw . . . . . . . . . . . . . . . . . . 31xFigure 3.2 FE half model (snippet showing mesh around the tip) of anOgden hyperelastic material (in grey) with symmetry about X-axis was developed to evaluate the energy release rate, J whenthe wire is inserted into a pre-existing crack. . . . . . . . . . 32Figure 3.3 (a) Semi-log plot of dimensionless energy release rate J/µRwvs δ/Rw for α = 2,5&9 (b) Log-log plot of dimensionlessstress σ22/µ vs dimensionless distance from the crack tip r/Rwevaluated when δ =Rw (blue), giving nonlinear elastic fracturemechanics (NLEFM) solution, and δ = 15Rw (δ ≫ Rw, red),giving linear elastic fracture mechanics (LEFM) solution . . . 33Figure 3.4 (a) Schematic showing the contact Pressure P and the criticalshear stress τc acting at the interface (b) Components of contactstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.5 Two representative scenarios for which we calculate dissipa-tive forces at steady-state Fd/ξµRwT as a function of dimen-sionless distance δ/Rw from the crack tip numerically for α =2, α = 5, and α = 9 . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.6 Log-log plot of dimensionless dissipative force at steady-stateFd/ξµRwT vs dimensionless cleavage toughness Γo/µRw atdifferent values of τw/µξ for α = 2, α = 5, and α = 9 usingABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.7 (a) Force per unit sample thickness (F/T ) vs Wire displace-ment (D) showing steady-state force per unit thickness (Fss/T )(b) Relationship between Fss/T (mean of five cutting tests) vswire radius Rw for a PAAm hydrogel, comparing experiments(blue squares) with a linear fit (red solid line), and with indi-cated Γ= 2.6 J/m2 (black dashed horizontal line). . . . . . . 43Figure 3.8 Relationship between Fss/T vs wire radius Rw highlightingcleavage toughness Γo (in a yellow cross) and R∗ (in a bluecross) which marks the transition to contact dissipation domi-nant effect making it less favorable than tearing limit estimatedfrom pure shear . . . . . . . . . . . . . . . . . . . . . . . . . 44xiFigure 4.1 (a) Semi-log plot of dimensionless energy release rate J/µRwvs δ/Rw for compressibility K/µ → ∞ (blue) and K/µ = 1(red) with strain stiffening parameters α = 2, α = 5, and α = 9(b) Log-log plot of dimensionless Cauchy stress σ22/µ vs di-mensionless distance from the crack tip r/Rw at δ = Rw forK/µ → ∞ (blue) and K/µ = 1 (red) with strain stiffening pa-rameters α = 2, α = 5, and α = 9 . . . . . . . . . . . . . . . 49Figure 4.2 Semi-log plot of (a) dimensionless Cauchy stress σ22/µ vsδ/Rw (b) dimensionless stress ratio σ22/σ11 vs δ/Rw . . . . 51Figure 4.3 (a) Semi-log plot of dimensionless deviatoric component ofCauchy stress σ2d/µ vs δ/Rw (d) dimensionless volumetriccomponent of Cauchy stress p/µ vs δ/Rw for compressibil-ity K/µ → ∞ (blue) and K/µ = 1 (red) with strain stiffeningparameters α = 2, α = 5, and α = 9 . . . . . . . . . . . . . 53Figure 4.4 (a) Dimensionless dissipative force Fd/ξµRwT vs dimension-less fracture toughness Γo/µRw (b) Dimensionless dissipativeforce Fd/ξµRwT vs dimensionless fracture toughness Γo/µRwfor α = 9 fitted and extrapolated with R2 ≥ 0.97 . . . . . . . 56Figure 4.5 Co-efficient m depicting shear sensitivity as a function of com-pressibility ratio µ/K . . . . . . . . . . . . . . . . . . . . . . 57Figure A.1 (a) Uniaxial Tension response of soft material σeng vs Stretch λ(b) Pure Shear INSTRON machine (c) Pure Shear response of atough soft material σeng vs Stretch λ for an Unnotched sample(red) and a Notched sample (blue) (d) Pure Shear response of abrittle soft material σeng vs Stretch λ for an Unnotched sample(red) and a Notched sample (blue) . . . . . . . . . . . . . . . 74xiiList of Symbolsr distance from crack tipRw Radius of the cutting wireD Indentation Depthδ Distance between the wire and the crack tipµ Shear modulusK Bulk modulusα Strain Stiffening parameterΓ Fracture EnergyΓo Intrinsic/Cleavage Fracture EnergyΓd Dissipative Fracture EnergyKI Stress Intensity FactorWf Strain energy density or Work of Fractureσ Cauchy stressλ Stretchε Small-strainl f Fracto-cohesive lengthscalele Elasto-cohesive lengthscaleF ForceFss Steady-state Cutting ForceFd Dissipative Cutting ForceP Contact Pressureτc Contact Shear Stressτw Shear resistance (adhesion/wear)ξ Friction CoefficientxiiiAcknowledgmentsFirst and foremost, I would like to thank my advisor Prof. Mattia Bacca for havingme work on this exciting problem. Additionally, I would like to thank Prof. DavidLabonte from Imperial College London, UK, for inspiring me with his work. Be-sides, I want to thank the members of the supervisory committee for their feedbackand other colleagues such as Dr. Frederik Puffel from Imperial College London,UK, Dr. Stefano Fregonese from the University of British Columbia, and Dr. Zhen-wei Ma, from the University of British Columbia.Finally, I would like to acknowledge the funding support of the Human Fron-tiers Science Program, the Department of National Defense (DND) of Canada, andthe Natural Sciences and Engineering Research Council of Canada (NSERC).xivDedicationI dedicate this thesis to my beloved parents, who have been my strength and sup-port. They always encouraged me to pursue my dreams which made all thesepossible. Moreover, I also want to thank and dedicate this to a special personfor showering me with genuine care and support while writing this thesis. Everyresearch is built upon previous studies, challenging previously held assumptions,filling gaps in understanding, and pushing the boundaries. I am greatly indebted toall the mechanicians who dedicated their lives to understanding the material worldwe inhabit.xvChapter 1IntroductionCutting is an important survival skill mastered by humans and animals. Some ofthe oldest cutting and piercing tools date back to the Stone Ages. Despite thelong-standing interest in this problem, we still lack a comprehensive mechanisticunderstanding of the rupture characteristic of soft materials subject to tearing andcutting. Understanding cutting is of enormous cultural and engineering importancewith applications ranging from food processing [1], manufacturing [2], and roboticsurgery [3]. We understand very little about cutting mechanics in soft materialsdespite the wide-ranging interest because of the complexity arising from a combi-nation of cutting tool geometry, contact conditions, and material properties [4]. Inparticular, the relationship between steady-state cutting mediated by contact loadsand concepts developed in fracture mechanics under remote loads. In this thesis,we explore the mechanism of cutting using the wire-cutting model. Typically, inwire cutting, we can identify three different stages (Figure 1.1) asi. Indentationii. Cut Initiationiii. Steady-state cuttingThis thesis focuses primarily on (iii) Steady-state cutting and the role of dissi-pative contact mechanisms such as friction, adhesion, and wear as well as materialproperties and constitutive behavior such as elastic-modulus, strain-stiffening, andcompressibility.1F(I)D(II)(III)Figure 1.1: Different Stages in Cutting (I) Indentation (II) Cut initiation (III)Steady-state1.1 OutlineThe objective of the thesis is to present a framework for understanding cutting,develop a quantitative relationship for cutting forces, accounting dissipative mech-anisms, and the effect of material properties such as toughness, stiffness, strain-stiffening, and compressibility. We first discuss the properties of soft materialsand then understand fracture mechanics and asymptotic stresses in soft and stiffmaterials. We revisit these concepts in the context of cutting. Then we incor-porate dissipative contact mechanisms and explore how they affect cutting forces.Chapter 1 introduces soft materials, discusses the role of dissipative mechanisms intoughness measured from remote tests, and explores the effects of network imper-fection. Section 1.3 discusses strain-stiffening, Section 1.4 reviews previous workson cutting, and is followed by Section 1.5 which explains the problem, the mainfocus of this thesis, and the methodology adopted. In Chapter 2, we present a dis-cussion on Linear and Nonlinear Elastic Fracture Mechanics valid for soft solids.Section 2.1 focuses on asymotic stress fields in LEFM and Section 2.2 introduces2two length scales, namely, fracto-cohesive and elasto-cohesive. Finally, Section3.3 discusses the implications of finite deformation at the crack tip using the ex-ample of Pure Shear and the associated stress fields. In Chapter 3, we discuss theCutting Mechanics of Soft Solids. Section 3.1 presents a theoretical and numericalframework to understand and predict dissipative forces due to frictional, wear, andadhesive mechanisms. Section 3.2 discusses experiments carried out to verify thepredictions. Section 3.3 discusses the results and explores different contact condi-tions: friction-dominated, adhesive/wear-dominated, and the complex combinationof both these dissipative mechanisms. Chapter 4 studies the effect of volumetriccompressibility on dissipative cutting forces. Finally, Chapter 5 presents the con-clusion, implications of the study, and recommendations for future work.1.2 Soft MaterialsSoft Materials are typically identified as materials with low stiffness or shear mod-ulus µ in the order of Pa to MPa (ex: Agar Gel or Tofu) whereas stiff or hardmaterials typically have shear modulus µ in the order of MPa to GPa (ex: Glass).Based on their inelastic behavior and fracture toughness Γ (in J/m2) which char-acterizes their resistance to crack propagation, we can further classify them Brittleand Ductile [5] (Figure 3.1).Figure 1.2: Material Classification based on stiffness: Siff/Hard or Soft, andbased on toughness: Brittle or Ductile [5]3Moreover, we note that additional classes of materials with discrete structures(meta-materials) [6] and Quasi-brittle materials [7] exist. However, in this thesis,we focus on soft brittle materials and understand the failure mechanisms undercontact loading (cutting), associated dissipative mechanisms (friction, adhesion,and/or wear) and explain the relationship with material properties such as stiffness(µ) and fracture energy (Γ).1.2.1 Fracture Mechanisms in Soft MaterialsThe fracture energy (Γ = Γo +Γd) in J/m2 of material provides a measure of itsability to resist fracture. Note that, Γo provides the intrinsic or cleavage fractureenergy, and Γd accounts for energy dissipated in the bulk. The stiffness (µ) in Pa isthe ability of the material to resist indentation at small strains. For soft brittle ma-terials (Γ≈ Γo), Lake and Thomas theory [8] (extended to hydrogels [5]) predictsintrinsic or cleavage fracture energy,Γo ∼ φ 2/3p Wf l√n (1.1)where n is the number of monomers, l is the length of a monomer making thepolymeric chain, Wf is the specific work of fracture in J/m3 or covalent energyper unit volume (in covalently bonded material), and φp is the polymer volumefraction in the swollen state. It is known that swelling reduces intrinsic fractureenergy because the polymer chains get stretched, and the critical failure stress (σ fin Pa) decreases [9]. In the case of an elastomer or an un-swollen gel (φp ≈ 1), wehaveΓo ∼Wf l√n (1.2)where l√n is mesh size (distance between crosslinks shown in Figure 1.3) typicallyin the order of few nanometers O(nm).4Figure 1.3: Mesh-like structure of an elastomer at nano-scales with cross-linking points denoted by x and the effective distance between thecrosslinks l√nWe can now define a lengthscale l f with dimensions of length called fracto-cohesive length scalel f ∼ Γo/Wf (1.3)Physically, the fracto-choesive length scale (l f ), characterizes the FractureProcess Zone (FPZ) [10].Note that, for an ideal-brittle soft material with a single network (SN), l f ∼ l√nand is O(nm) as shown by Lake and Thomas [8] (Figure 1.4 (i)). We now need tounderstand the role of chain length on stiffness µ and its relationship with thresh-old or cleavage toughness Γo.Now, an increase in monomer concentration (n) or shortening of chain length(l√n), stiffens the materialµ ∼ 1/n (1.4)but embrittle themΓo ∼√n (1.5)This informs the conflict between stiffness and intrinsic toughness in soft materials[11].5Figure 1.4: Different dissipative mechanisms in Soft Solids with FractureProcess Zone (FPZ) highlighted around the crack tip in grey and thetypical size of FPZ marked below (source: [5])In the case of a Double Network (DN) hydrogel [12], to overcome the stiffness-threshold conflict noted in Eq. (1.4) and Eq. (1.5), two networks of differentmaterials and chain lengths are introduced givingµ ∼ 1/nshort (1.6)Γo ∼√nlong; Γd ∼√nshort (1.7)For a DN hydrogel, the long chain network provides high intrinsic toughnessΓo as previously noted. However, the short-chain network gives high stiffness anddissipative toughness Γd and acts as sacrificial networks that resist fracture propa-gation giving a large fracture process zone l f around the crack tip in the O(mm).Like in a DN gel, dissipative mechanisms can be introduced using multiple cross-linking mechanisms, involving physical entanglements and chemical bonding suchas in PAAm-Alg gel. Other dissipative mechanisms can be introduced into these6soft synthetic materials at multi-scales in the form of crystallization and fiber rein-forcement as summarized in [13]. Notably, these mechanisms are also identified inbiomaterials [14].Network ImperfectionsThe focus of this thesis is on a single network (SN), soft brittle materials suchas Polyacrylamide (PAAm). Interestingly, although soft and brittle, PAAm hasl f or FPZ in O(mm) despite the absence of extrinsic mechanisms by which theydissipate. The physical mechanisms governing dissipation in these materials wereidentified and explained to be due to network imperfection [15] and illustrated inFigure 1.4 (ii).Network imperfections in the material manifest themselves by causing somechains in the material to be shorter than the main network. Now, when such amaterial is subject to a remote load, it results in premature scission of the shorterchains (due to imperfections) in the bulk. Thereby, it effectively shields the cracktip or scales the apparent toughness Γ = Γo +Γd or Γ ∼ Γd by having a large Γd .Therefore the toughness Γ measured from conventional tests (such as Pure Shear),can be larger than cleavage toughness Γo (a material property) when there are moreimperfections or relatively shorter chains in the material [15]. Therefore, we havethe following conditionΓ= Γideal ≈ Γo (Perfect Network) (1.8)Γ> Γideal (or) Γ= Γo +Γd (Imperfect Network) (1.9)Now, the work of fracture (Wf ) in an imperfect material decreases from theideal theoretical value Wideal estimated for a perfect material [15]. We thereforehave Wf < Wideal and using the condition in Eq. (1.9) in estimating l f from Eq.(1.3), we getΓo/Wideal ≪ Γ/Wf (1.10)7I.e., we have the condition for FPZ size l√n≪ l f . This explains the scaling ofFPZ from O(nm) in an ideal material to O(mm) in the presence of imperfections.This suggests why the FPZ size, estimated from remote fracture tests will be largerthan the ideal mesh size.1.3 Strain-StiffeningSoft materials both synthetic and biological may exhibit strain stiffening [16]. Sev-eral different constitutive models, both phenomenological [17] and statistical [18]variants, can capture the nonlinear stress-strain relationship exhibited by these ma-terials. The most commonly used model is the Ogden hyperelastic model with theparameter α capturing the strain-stiffening tendency in the material [19]. Now,to motivate the role of strain stiffening, let us consider an incompressible Ogdenhyperelastic strain energy density (W ) given byW =µα(λα1 +λα2 +λα3 −3) (1.11)where λi are principal stretches and α characterizes strain stiffening tendency ofthe material [20]. When α = 2, we haveW =µ2(tr(C)−3); tr(C) = λiλi (1.12)is the strain energy density functional for Neo-Hookean hyperelasticity with theconstraint λ1λ2λ3 = 1 due to incompressibility.When a pristine, uniaxial test specimen is stretched by λ in the vertical di-rection X2, we have the following condition for principal stretches: λ2 = λ &λ1 = λ3 = 1/√λ . The critical strain energy density Wf (λ ,α) and the failure stressσ f (λ ,α) at which the sample fails catastrophically are given byWut =∫ λc1σ22dλWf (λ ,α) =µα(λα +2λ−α/2−3) (1.13)σ f (λ ,α) = µ(λα −λ−α/2) (1.14)8For a given fracture energy Γ ∼Wf l f , i.e., when critical strain energy densityWf (λα=2c ,2) =Wf (λα=5c ,2), from Eq. (3), we see thatλα=2c ≫ λα=5c (1.15)using Eq. (1.14) and (1.15), we getσ f (λα=2c ,2)≪ σ f (λα=5c ,5) (1.16)we note that the critical strain energy density is attained at a lower stretch byeffectively localizing the stresses at a higher α despite having the same ground stateshear modulus (µ) i.e. with the same initial slope µ in σ vs λ curve. Effectively,strain-stiffening α becomes important at large stretches (Figure 1.5).Figure 1.5: (a) Dimensionless Cauchy stress σ22/µ vs remote stretch λ2 (b)Dimensionless strain energy density Wf /µ vs remote stretch λ291.4 Problem DescriptionOver the past few decades, much attention has been paid to understanding soft ma-terial failure under remote loading conditions (pure shear, tensile test, etc.) andonly a few studies have investigated in detail the cutting mechanics of soft materi-als. As a consequence of entropy-driven elasticity, soft solids such as elastomers,and hydrogels undergo large deformations when subject to contact forces. At acritical threshold, the material undergoes fracture nucleation followed by propaga-tion (steady-state) as discussed previously. The goal is to identify the relationshipbetween concepts developed in the fracture of soft solids to cutting.Figure 1.6: Relationship between cutting and fracture10Much progress has been made in understanding the elastic response of stiff ma-terials based on small strain kinematics. Moreover, inspecting the tensile, cyclic,or tri-axial compression response of stiff-ductile materials (eg., metals, rocks, etc)to failure, helped us understand inelasticity in the form of plasticity: pressurein-dependent (Tresca, Von Mises) and pressure-dependent yield (Mohr-Coulomb,Drucker Prager) criteria, hardening laws, and a rigorous investigation on their mi-croscopic origins [21]. In the context of fracture, this motivated the extension ofGriffith’s theory [22] to account for plastic dissipation in the process zone [23].Further, the role of in-elasticity in the fracture of stiff materials and its effect onasymptotic stresses was studied extensievly by Hutchinson-Rice-Rosenberg ([24].However, we still lack a thorough understanding of soft solid fracture and the fail-ure of these materials under different loading modes such as contact (cutting, punc-ture, etc) and the role of contact dissipation.We understand this from the framework of wire cutting shown in Figure 1.7(a) where the wire in (brown) is pushed into a soft material (in green) of length Land thicknesses T. The force (F) needed to grow the cut is recorded as a functionof displacement (D) in X1 and reported in Figure 1.7 (b). This 1.7 (b) shows allthe regimes of cutting discussed previously. In particular, the steady-state forcereported in the figure is Fss ≈ 0.4N. However, the following questions arise:• What factors affect the magnitude of steady-state force?• Is steady-state force per unit thickness Fss/T , a material constant indepen-dent of cutting tool dimensions?• What is the role of the cutting tool radius?• Effects of contact dissipation due to friction, adhesion, and damage (wear)• Role of strain stiffening tendency and compressibility of a soft material11Figure 1.7: (a) Schematics of Wire Cutting (b) Force (F) vs IndentationDepth (D) response showing different stages in cutting1.5 Literature ReviewThe rise of industrial applications for soft materials such as rubber in the 20th cen-tury, saw several studies dedicated to understanding its failure characteristics. Sig-nificant efforts were taken to improve the toughness of rubber and other elastomericmaterials [25]. Rivlin, Lake, Thomas, and several other coworkers published a se-ries of works to understand the failure behavior of these materials under differentexperimental setups: Pure Shear [26], Tensile [27]. Different loading conditions:Cyclic fatigue [28, 29] and Cutting resistance [30, 31] of these nonlinear elasticmaterials in the absence of contact dissipation.Notably, to understand steady-state cutting, some earlier works on rubber [30]and more recent work that focus on other soft elastomers and gels [32, 33] soughtto understand the relationship between the cutting tool radius and fracture energyby designing thin Y-shaped samples (shown in Figure 1.8) and studied cutting atsteady-state conditions.This setup was designed with the aim of controlling or minimizing friction incutting tests by reducing the interfacial contact area i.e., by changing the boundary12FFigure 1.8: (a) Schematics of Y-shaped cutting setup (b) Striation in the cutsurface in the far-field moderate or high tearing regime (source: [30])conditions at the crack tip, as shown in the setup sketched in Figure 1.8 (a). Thiswas done by tuning the opening angle (θ ) of the Y-shaped sample (in grey) withrespect to the cutting tool (in red). They note that, with an increase in θ , theyobserved a transition from the True Cutting regime to the Tear Dominant far-fieldloading condition. In the True cutting regime, the indentation of the crack tip domi-nates the cutting process, while in the latter the crack propagates at a distance awayfrom the tip of the cutting tool. Moreover, True-cutting regime was identified asa region with a clean continuous cut whereas the far-field tear-dominated regimewas identified with stick-slip cutting [30, 32] with rougher cut surfaces (Figure 1.8(b)).From our everyday experience, we have an intuitive understanding that brittlematerials would require a lower force to propagate a cut. However, as demon-strated and explained by Spagnoli et al., the cutting response also depends on thestiffness of the material [34, 35], cutting angle [36], and shearing [37]. Some otherworks that focus on these frictional instabilities and failure in soft solid contact dueto shearing include the works in [38–40]. Figure 1.9 shows the cutting schematicsadopted in [34] under steady-state conditions (ignoring out-of-plane effects). Theyconsider a stiff-brittle material such as a polymeric glass (in grey) cut with a blade(in black) to propagate a crack of length a. For a brittle material such as polymeric13Figure 1.9: (a) Cutting hard and soft solids (in grey) by blade (in black)(source: [5]) (b) Physical picture capturing the location of crack tip inrelation to the blade tip while propagating the cut (source: [5])glass (with high stiffness µ), they note that the displacement of blade tip D is lower.We then have the condition D≪ a, i.e., a cut or crack tip propagating at a distanceaway from the blade tip (see Figure 3.3 (b)). In this regime, the geometry of the tipdoes not mediate the cut propagation [34]. However, when a silica elastomer witha low stiffness (µ) is cut with a blade, they notice significant crack-tip deformation[34]. While useful, this study does not provide a full picture and does not providea quantitative relationship for the forces accounting for a complex combination ofdissipative mechanisms. Further, it does not investigate the role of strain-stiffeningα and compressibility K/µ typically observed in soft materials.This motivates a careful consideration of Γ in relation to µ , Wf , α and K/µ incutting. We can understand this by introducing two length scales that characterizefracto-cohesivity: l f ∼ Γ/Wf and elasto-cohesivity: le ∼ Γ/µ [5] (see Section 2.3).In addition, we need to understand the effect of strain-stiffening or bulk modulusis not accounted for in these lengthscales. It is, therefore, important to understandthe role of these length scales in remote loading scenarios and explain its impli-cations in the context of wire cutting which introduces a wire of radius (Rw) asthe geometric parameter. Finally, we need to understand also the role of contact14dissipation in the form of adhesion and friction and provide a quantitative measureof the steady-state cutting forces required to propagate a pre-existing cut.Using carefully designed experiments for validation, we need to explain howone can measure properties of materials such as fracture energy (Γ), cleavagetoughness (Γo), and sensitivity to wire radius (Rw) under different cutting con-ditions. This would help illustrate that, by understanding the cutting mechanics ofsoft materials, we can gain insight into ways to develop tougher materials that areharder to cut or help identify best-cutting tool design strategies and contact condi-tions that facilitate cutting.15Chapter 2Fracture Mechanics2.1 Linear Elastic Fracture Mechanics2.1.1 K-fieldThe well-known stress and displacement fields in LEFM valid in the region nearthe crack tip are [41]σi j =Ki√2πrfi j(φ)+H.O.T (2.1)ui =Ki2E√r2πfi(φ)+H.O.T (2.2)where E (in Pa) is the Young’s modulus, Ki (in Pa√m) is the stress intensity factorin mode i = 1 to 3, fi j and fi are dimensionless functions of φ , the inclination withrespect to the crack plane (φ = 0 implies ahead of crack tip) as shown in Figure 2.1We choose the reference configuration to be the undeformed, stress-free contin-uum body with a material point identified by (X1,X2). Upon loading, the materialpoints in the deformed configuration are identified by (x1,x2).Mode I is defined by requiring the displacement fields to be symmetric about162cLLFPZK fieldcrFigure 2.1: Square plate of Length (L), with a center crack of length (2c),such that L ≫ c, subject to a remote load σ∞ in X2. Polar coordinates(r,φ ) around the crack tip (Right)the crack line, i.e.,u1(X1,X2) = u1(X1,−X2), u2(X1,X2) =−u2(X1,−X2) (2.3)Here, the stress field is symmetric with respect to the crack plane, which occurswhen both the specimen geometry and the applied loads are symmetric. Therefore,there are no shear stresses on the crack extension line, no relative slip along thecrack face and crack parallel stresses are assumed negligible. This mode of fractureis called mode I, or the opening mode. In mode II, also called the in-plane shearmode, the planar stress field is anti-symmetric with respect to the crack plane. Inmode III, also called the anti-plane shear mode, the displacements are asymmetric17in the X3 out-of-plane direction and shear stresses τ13 and τ23 exist on the crackplane. [41, 42]. LEFM assumes these modes are separable and can be superim-posed to explain the complex combination of all three modes. However, in the caseof Mode-I loading (our focus), which is the low-energy mode, we have differentcomponents of Cauchy stresses given byσ11σ22σ12= KI√2πrcosφ21− sinφ2 sin3φ21+ sinφ2 sin3φ2sinφ2 cos3φ2+H.O.T (2.4)It is this K we consider to be the singular parameter characterizing the stressesand defined in terms of the remote stress σ∞ and crack size 2c, asKI = σ∞√πc (2.5)using Eq (2.4), for φ = 0o, for the normal component of stress in X2, we haveσ22 =KI√2πr(2.6)the well-known square-root stress singularity as r → 0.Now, we use ABAQUS 2022, to compute the stress fields around the crack tipin the square plate shown in Figure 2.1 (Right) for the case where the remote stressis σ∞ = 100 Pa and a crack length of 10 mm. To perform the analysis, 4 nodedlinear continuum plane stress elements (CPS4) were used to discretize the model,and a static-general implicit procedure was adopted to ignore the inertial terms inbalance laws (no rate effects). Remote stress was applied as shown in Figure 2.1along X2.Figure 2.2 (a), plots the results of the numerical model (in blue) for Cauchystress σ22 along the distance from crack tip r at φ = 0o along with the valuescalculated from Eq. (2.6) as the analytical solution (in red). We note that, closer tothe crack tip, we have a region of stress singularity and the K-field zone is valid inthe region slightly away from the crack tip. Far away from the crack tip, we haveσ22 → σ∞ (100 Pa in our case), i.e., at a distance away from the crack tip, stressesare unaffected by the singular parameter KI and linear elasticity holds.1821Figure 2.2: (a) Plot of Cauchy stress (σ22) in X2 as a function of distancefrom the crack tip (r) at φ = 0o (b) Log-log plot of σ22 vs r highlightingthe validity of K-field in the region close to the crack tip.19Note that, for the in-plane stresses in Eq (2.4), in plane-strain conditions, theout-of-plane stress σ33, for an in-compressible solid (Poisson’s ratio, ν = 0.5), wehaveσ33 = ν(σ11 +σ22) =12(σ11 +σ22) (2.7)σ33 =KI√2πrcosφ2+H.O.T (2.8)Now, along the length of the crack, we have φ = 0o. Using Eq (2.4), we havestresses in all three directions ahead of the crack tip in an in-compressible materialas,σ11 = σ22 = σ33 =KI√2πr(2.9)which explains that closer to the crack tip, we have a region of stress-triaxialitymarked by the K-field zone in LEFM for an incompressible material.2.2 Fracto-cohesive and Elasto-cohesive Length ScalesThe previous section discussed the K-field using an example of an infinite platewith a center crack. More generally, we haveKI = Yσ∞√πc (2.10)where Y is the shape factor (or) KI ∼ σ∞√c (dropping coefficients). Now, we havethe following conditions for crack growth• If KI < KIc then no crack growth (stable equilibrium due to stress deficit)• If KI ≥ KIc Quasi-static or catastrophic crack-growthwhere KIc is the critical stress intensity factor (SIF) or fracture toughness and is amaterial property. SIF can be related to cleavage fracture energy, Γo (in J/m2) or20generally, Γ ≈ Γo (in brittle material), and Young’s modulus, E (in Pa) using thewell-known relationshipKIc =√ΓE (2.11)Here, KI = KIc under following conditions:(i) Presence of a flaw: where c crack/flaw length in Eq. (2.10). Now, using Eq(2.11) and (2.10) we getΓ∼ σ2∞E(c) (2.12)(ii) Absence of a flaw: when σ∞ = σc where σc is the cohesive strength (amaterial property) and we haveΓ∼ σ2cE(l f ) (2.13)Here l f (with the dimensions of length) is in the order of atomic bond length for anideal-brittle material such as glass [10]. More precisely, the area under the stress-strain curve of an un-notched sample or Work of fracture Wf (in J/m3), as shownin Figure 2.3 (a) and (b), is given asWf =12σcε =12σ2cE(Linear Elastic σ = Eε) (2.14)Wf ∼ σ2cE(2.15)Now, using Eq. (2.13) and Eq. (2.15), we haveΓ∼Wf l f (or) (2.16)l f ∼ ΓWf (fracto-cohesive) (2.17)Physically, the fracto-cohesive length scale (Eq. (2.17)) represents the dissipa-tive fracture process zone (FPZ) around the crack tip. In the case of an ideal brittlematerial such as Silica glass, FPZ or l f is idealized to be in the order of nanometers21(nm) or atomic bond length (Figure 2.3). In the order of micrometers (µm) in met-als, and in the order of millimeters (mm) for soft elastomers and hydrogels (Figure2.3) [10] [15].Figure 2.3: (a) Typical Stress vs Strain response of a Linear-Elastic mate-rial with work of fracture Wf (area under the curve) shaded in blue(b) Typical Stress vs Strain response of a Nonlinear-Elastic material (c)Fracture Energy and Work of fracture/rupture Ashby plot (source: [10])(d) Showing crack-opening profile of soft material with a critical crackopening displacement δctod = 7mm (source: [5])Now, by revisiting the crack opening profile from Eq. (2.2) (shown in Figure2.3), we have u(r) =∫ε(r)dr (or) u(r) ∼ KI√r/E. Where the critical crack tipdisplacement u(x) where x is coordinate, and the critical radius of curvature of the22tip ρc areu(x)∼ KIc√xE(2.18)ρc ∼ 1/δ2xδu2∼ K2IcE2∼ ΓE(2.19)le ∼ ΓE (elasto-cohesive) (2.20)Using, Eq. (2.17) and Eq. (2.20) give us two important length-scales in fracturemechanics with l f (fracto-cohesive lengthscale) characterizing the FPZ around thecrack tip and le (elasto-cohesive lengthscale) [43] characterizing the deformationsaround the crack tip before failure.2.3 Nonlinear Elastic Fracture MechanicsSo far, in section 2.1, we discussed the role of the singular parameter KI and thevalidity of the K-field zone around the crack tip. In particular, as shown in Eq(2.9), in LEFM, for an incompressible material (ν = 0.5), we have a state of stresstriaxiality in this region. Much closer to the crack tip, we have the Fracture ProcessZone (FPZ) characterized by l f as discussed in section 2.2. This is the pictureprovided by LEFM and depicted in Figure 2.4 (a).FPZK fieldK fieldcInner NonlinearElastic ZoneFPZcFigure 2.4: (a) FPZ and K-field zone around crack tip in LEFM (b) FPZ,Nonlinear Elastic zone, and K-field zone around crack tip in NLEFMTypically, LEFM is valid for stiff brittle materials with a very high stiffness µ23which gives µ ≫Wf (Eq. 2.15) for a given Γ [5]. However, in the case of softmaterials (low µ), we typically have the condition µ ≪Wf for the same Γ. I.e., ata given Γ and µ , the area under the Stress-Strain/Stretch curve Wf is larger for softmaterial because soft materials experience finite deformations at the crack tip priorto failure [5]. Therefore, we note the following conditionΓµ≪ ΓWf(Stiff Material) (2.21)Γµ≫ ΓWf(Soft Material) (2.22)From Eq. (2.21), for a siff material (in LEFM), we see that l f (fracto-cohesivity)is larger than le (elasto-cohesivity). That is, le characterizing crack tip deformationis contained within the FPZ l f . However, for a soft material (in NLEFM), we havel f ≪ le and therefore, the zone of nonlinear elasticity or crack tip curvature char-acterized by le is significant and cannot be ignored. Therefore, in addition to FPZ,and the K-field, we have Inner Nonlinear Elastic zone characterizing finite defor-mation at the crack tip of a soft material where µ ≪Wf (Figure 2.4(b)). Despitebeing brittle and having the same fracture energy (low Γ), the consequence of mag-nitude of stiffness µ , in relation to work of fracture Wf , and how it affects the stressstate around the crack tip is explored in section 2.3.1 using the example of a PureShear setup.2.3.1 Pure ShearWe consider an incompressible Ogden hyperelastic strain energy density (W ) de-fined asW =2µα2(λα1 +λα2 +λα3 −3) (2.23)where λi are principal stretches and α characterizes strain stiffening tendency ofthe material [20]. W depends on µ , λ , and α , with µ = NkT the shear modulus,N the cross-link density, k Boltzmann’s constant, and T the absolute temperature.Note that α = 2 gives neo-Hookean hyperelasticity and higher α represents strain-24stiffening typical of many synthetic and biological materials.In the case of pure shear fracture tests (Figure. 2.5), Rivlin et al [26] extendedGriffith theory [22] from linear elastic materials to soft hyperelastic materials likerubber, providing the fracture energy asΓ=WpsH (2.24)whereWps =2µα2(λαps +λ−αps −2) (2.25)is the pure shear strain energy density, with λ2 = λps the pure shear stretch in X2 onthe sample with an undeformed height H. Eq. (2.25) was obtained by noting that,in pure shear, λ1 ≈ 1, using the incompressibility condition, we have λ3 = 1/λ2 =λ−1ps . Here,σps =2µα(λαps−λ−αps ) (2.26)is the Cauchy stress (in X2) in the pure shear region maintained away from thecrack tip due to the remote stretch [26]. From LEFM, we note that the stresses nearthe crack tip shown in Figure.2.5 areσ11 = σ22 = σ33 ∼ KI√r (2.27)where KI is the Mode-I Stress Intensity Factor (SIF), and r is the distance from thecrack tip in the direction of the crack plane. Eq. (2.27) describes the stress state inthe K field around the crack tip and highlights stress triaxiality.However, as noted in [41], in a soft hyperelastic material (µ≪Wf ), the K fielddescribed by Eq. (2.27) is valid only in the small region outside the NonlinearElastic Zone (NEZ). In the NEZ, the material experiences significant nonlinearelastic deformation due to stress uniaxiality Figure 2.4. In this region, the stressstate is [5, 41]σ22 ∼ Γr ;σ11σ22,σ33σ22≈ 0 (2.28)25identifying the stress field in non-linear elastic fracture mechanics (NLEFM).To understand the role of strain stiffening in this problem, let us apply a remotestretch λps = 3 in X2 on the Ogden material shown in Figure 2.5. We haveλps|α=2 = λps|α=5 = 3 (2.29)Now, from Eq. (2.26) & (2.29) we obtain the condition on the stress σps in X2σps|α=2 < σps|α=5 (2.30)rFigure 2.5: (a) Pure Shear sample of Length (L) and Height (H) with an as-pect ratio (L/H = 100) and crack (in red) of length (c = 0.4L) snippetshowing the mesh around the Crack tip (b) Results of a simulation ofNeo-Hookean sample with a remote stretch in X2 of λ2 = 326First, we note that, far away from the crack tip r≫H, the K-field from fracturemechanics is invalid and we have the Pure Shear zone [26]. In the pure shear zone,we have the stresses σps|α=2/µ = 8.89 (blue) and σps|α=5/µ = 97.19 (red). Thiscan be verified by evaluating these for λps = 3 in Eq. (2.27). We can see thisin Figure 2.6 (a) for remote stresses corresponding to remote stretch approachingλ2 ≈ 3 far away from the tip (r ≫ H) Figure 2.6 (b) for both α = 2 (blue) andα = 5 (red). This verifies that the analysis is self-consistent. In particular, we notethat, at a given stretch, α = 5 attains higher global remote stresses σps comparedto α = 2 as seen in Eq. (2.30).Now, as we approach closer to the crack tip, in pure shear, we have KI ∼σps√Hand σ22∼KI/√r. Now, we get σ22∼σps√H/r (or) σ22/µ ∼√H/r. I.e., as notedin Eq. (2.27), we have the well-known square-root singularity r−1/2 for stresses.This is noted in the stress field for α = 2 (blue) and α = 5 (red) but contrary toLEFM, this singularity is confined only to a small region as we approach the cracktip.Further closer to the tip r ≪ H (right to left in Figure 2.6 (a)), we see the tran-sition from K field zone to NEZ, i.e., from Eq. (2.27) to (2.28) for the stress field(see (Figure 2.6 (a)). This transition highlights the boundary of the NEZ whichmarks the region with singularity r−1 dominated by nonlinear deformations andstress uniaxiality irrespective of α . I.e., we note that we have σ22/µ ∼ H/r forα = 2 (blue) and α = 5 (red).Notably, we observe that the transition point is closer to the crack tip for higherα . This is because the stress σ22 approaches its pure shear value (σps = 8.89 (blue)and σps = 97.19 (red) ) due to global stretch at r ≫ H, and σps changes with α , asstated in Eq. (2.30). Thus, with higher α , the stress approaches its (higher) pureshear stress state at a shorter distance from the crack tip. For example, considera critical value of strain energy density Γ/µH ∼Wps/µ = 3.56. The correspond-ing critical remote stretches in this case are λps|α=2 = 3 and λps|α=5 = 2.15. Thestresses are σps|α=2/µ = 8.89 and σps|α=5/µ = 18.56. This illustrates that, at agiven Γ, Wf and µ , α = 2 attains a higher degree of stretch uniaxiality or crack tipdeformation/blunting before failure.272112111Figure 2.6: (a) Dimensionless Stress field (σ22/µ) evaluated near the cracktip of an Ogden hyperelastic material (α = 2,5) in Pure Shear using FEsimulations (b) Stretch field (λ2) evaluated near the crack tip of an Og-den hyperelastic material (α = 2,5) in Pure Shear using FE simulations.28Moreover, this interpretation is further verified by plotting the asymptotic stretchesλ in X2 see Figure 2.6 (b). At r ≫ H, λ2 approaches the remote stretch λps = 3irrespective of α . At r ≪ H, the scaling relation for stretch λ2 ∼ (r/H)−1/α , sug-gests a reduced crack blunting in highly strain-stiffening materials (high α). We,therefore, conclude that α determines the size of the NEZ, and thus the regiondominated by stress uniaxiality and the extent of crack-tip deformation at failure.The elasto-cohesive length le ∼ Γ/µ does not capture the effect of α at a givenΓ and µ . From this simple problem of pure shear, we note that lower α = 2 effec-tively deconcentrates the stresses by exhibiting a higher degree of stress uniaxialitycausing a larger crack-tip stretch than α = 5 which stiffens when stretched.Understanding the asymptotic behavior is of great importance because the stressstate at the crack tip exerts a fundamental influence on the mechanisms of failureat the continuum scale thereby dictating the conditions of crack propagation. Thisin cutting, can be useful in explaining how they influence dissipative mechanismsthat are sensitive to contact pressure and shear stresses at failure for a given Γ andµ . Notably, how these parameters affect the cutting forces will be explored in thenext chapter.29Chapter 3Cutting MechanicsCutting involves the propagation of a cut due to contact loading. In the example ofwire cutting shown in Figure 3.1, crack propagation is mediated by a cutting tool(the cylindrical wire), which loads the sample via contact stresses in the proximityof the crack (cut). This loading condition is different from remote loading, like thepure shear test.Let us now analyze the steady-state cutting force Fss required to continuouslycut the sample by pushing the wire through it in X1 (Figure 3.1). Consider the softmaterial specimen (in grey) of length L and thickness T with a pre-existing cutof depth c (in red), in which we insert the rigid wire of cross-sectional radius Rw(Figure 3.1). Now, the work of cutting, per unit depth D in X1 direction, isFssdD = ΓoT dc+dUs +dQd (3.1)where Γ0 is the toughness of the material Us is the stored elastic strain energyin the sample, and Qd is an additional mechanical dissipation energy, besides theenergy required for crack propagation. We account for Qd as the work done by thedissipative forces like friction, adhesion, and wear.For example, when dD = 0 we can neglect dQd as all dissipation mechanismsare activated only in the presence of wire motion. Thus, we haveΓo =− dUsT dc (3.2)30i.e., the elastic strain energy accumulated in the sample is immediately released viacrack propagation, following Griffith’s law.Figure 3.1: Schematics of the setup adopted to perform wire cutting along X1on the soft hyperelastic sample (in grey) by a cylindrical wire (in brown)of radius RwNow, at steady-state dc = dD, in the absence of dissipation, we haveFss = ΓoT (3.3)Under these conditions, we note that the cutting force is independent of the cuttingtool radius Rw. From Figure 3.1 and Figure 3.2, we see that δ is the distance fromthe center of the wire of radius Rw to the crack tip (marked as X in red in Figure3.2). Therefore, the steady-state cutting configuration is given by dimensionlessdistance from the crack tip δ/Rw. When the wire touches the crack tip at steady-state, we have δ/Rw = 1. However, when cutting occurs at a distance from the tip,31Figure 3.2: FE half model (snippet showing mesh around the tip) of an Og-den hyperelastic material (in grey) with symmetry about X-axis was de-veloped to evaluate the energy release rate, J when the wire is insertedinto a pre-existing crack.we have δ/Rw > 1.Now, to understand the relationship between energy release rate J, µ and cut-ting configuration at steady-state δ/Rw, we perform VCCT (virtual crack closuretechnique [44]) simulations on ABAQUS using R2D2 (a 2-node 2-D linear rigidlink) for the cutting wire and CPE4H (a 4-node bilinear plane strain quadrilat-eral, hybrid, constant pressure) element to handle material incompressibility of thesample (Figure 3.2). To estimate the energy release rate, a crack was assigned tothe symmetry plane as an engineering feature with a crack extension direction (q-vector). Using FEA, we obtain the following relationship between dimensionlessenergy release rate J/µRw and cutting configuration δ/RwJµRw∼(Rwδ)2(3.4)as plotted in Figure 3.3 holds irrespective of strain-stiffening α .32Figure 3.3: (a) Semi-log plot of dimensionless energy release rate J/µRw vsδ/Rw for α = 2,5&9 (b) Log-log plot of dimensionless stress σ22/µ vsdimensionless distance from the crack tip r/Rw evaluated when δ = Rw(blue), giving nonlinear elastic fracture mechanics (NLEFM) solution,and δ = 15Rw (δ ≫ Rw, red), giving linear elastic fracture mechanics(LEFM) solution33In Figure 3.3, for δ > Rw, i.e., J < µRw, a regime that we call far-field cutting,while for δ ≈ Rw we have J ≫ µRw, in the indentation-cutting regime, i.e., whenthe wire touches the crack tip during propagation.Adopting J = Γo, when Γo ≫ µRw, the wire fully indents the crack at steady-state. We call this the True Cutting region where Γo/µRw is high and δ/Rw = 1 dueto crack tip indentation (Figure 3.3 (a)). In the far-field cutting regime, we observea stress concentration that is akin to that from linear elastic fracture mechanics(LEFM), as shown in Fig. 3.3(b) from our FEA, with δ = 15Rw. In the indentation-cutting regime, on the other hand, we observe a stress concentration that is akin tothat from nonlinear elastic fracture mechanics (NLEFM) [5], as shown in the samefigure, from our FEA with δ = Rw. In both cases, we see that the Pure Shearzone: σ22 = µ(λα + λ−α) is not present in wire cutting due to the absence ofremote loading. Therefore, σ22/µ → 0 far away from the crack tip. Under theseconditions, the cut propagation is steady (i.e., not catastrophic) and is a commonfeature in soft material cutting.The increase in energy required to propagate the cut at high Γo/µRw causesan increase in the contact pressure, P, and is marked by high-stress uniaxialityi.e., tensile in X2 (Figure 3.2). Now, by imposing kinematic constraints of a finiteincompressible material (J = λ1λ2λ3 = 1), we getλ2 = 1/λ1; λ3 = 1 (plane strain) (3.5)σ22 ≫ σ11 when λ2 ≫ λ1; σ22 ∼ 1/r (3.6)where λ1 is the compressive stretch in X1 due to indentation and the degree of stressuniaxiality increases with Γo/µRw or at δ = Rw. In this regime, we see a deep in-dentation of the crack tip by the cutting tool.The elasto-cohesive length scale: le ∼ Γo/µ which marks the transition fromstress triaxiality to stress uniaxiality in Pure Shear test (Figure 3.1(b)). Here, thismarks the transition from to Far-field Cutting (low toughness or high stiffness orlarge radii) marked by low crack tip indentation to True Cutting (high toughness orlow stiffness or small radii) regime marked by significant crack tip indentation.343.1 Role of friction, adhesion and wearFrom Eq. (3.3), we note that the force needed to advance the crack in X1 is Fss =ΓoT . Now, in the presence of dissipative forces, Fd , we haveFss = ΓoT +Fd (3.7)where Fd = dQ/dD is the resultant of all the dissipative forces projected alongthe wire path or cutting direction X1.Figure 3.4: (a) Schematic showing the contact Pressure P and the criticalshear stress τc acting at the interface (b) Components of contact stresses35The dissipative force Fd depends on interfacial contact interactions between thewire and the specimen such as friction, adhesion, and wear. Fd is then the integralof the projection of the (interfacial) contact shear stress τc along the X1 direction(see Figure 3.4(b)), givingFd = 2Rw T∫ θ2θ1τc sinθdθ (3.8)withτc = ξP+ τw (3.9)as shown in Figure 3.4(a) and Figure 3.4(b). Here, ξP is the frictional shear stresstransmitted at the interface, with ξ the friction coefficient and P the contact pres-sure. τw is the critical shear stress characterizing both the adhesive strength be-tween the wire-specimen interface and wear. Therefore, the Eq. 3.9 is a modifiedCoulomb friction model [45].Using Eq.(3.9) in Eq. (3.8), Fd/RwT can be decomposed intoFdRwT= 2τw∫ θ2θ1sinθ dθ +2ξ∫ θ2θ1Psinθ dθ (3.10)where θ1 and θ2 are the angles at which we have zero contact pressure (P(θ1) =P(θ2) = 0). By rearranging Eq. (3.10) we haveFdRwT= 2τw (cosθ1− cosθ2)+2ξ FhRwT (3.11)where Fh =∫ θ2θ1 Psinθ RwT dθ is the horizontal force applied to the wire to push itinto the half-specimen in X2, and ξFh is the frictional resistance projected in X1 asshown in Figure 3.4 (b).Using FEA, we calculate θ1, θ2 (shown in Figure 3.4), and Fh, as functions ofthe configuration of the wire-specimen system, which is given by the ratio δ/Rw toobtain the dimensionless dissipative force by dividing Eq. (3.11) by µξFdξµRwT= 2τwµξ(cosθ1− cosθ2)+2 FhµRwT (3.12)36The procedure adopted to obtain the relationship between dimensionless dis-sipative force Fd/ξµRwT and dimensionless fracture energy Γo/µRwT is as fol-lows: We know the relationship between dimensionless fracture energy Γo/µRwTand configuration of the cutting wire δ/Rw as shown in Figure 3.3 (a). Now, weestimate Fh/µRwT in Eq. (3.12) at different configurations (see Figure 3.5). Fi-nally, we vary τw/µξ from 0 to 100 and plot the dimensionless dissipative forceFd/ξµRwT as a function of Γo/µRw for α = 2, 5 and 9.Case 1 Case 2Figure 3.5: Two representative scenarios for which we calculate dissipativeforces at steady-state Fd/ξµRwT as a function of dimensionless dis-tance δ/Rw from the crack tip numerically for α = 2, α = 5, and α = 9We now have the relationship between Fd/ξµRwT at different δ/Rw. Because wehave identified relationship between Γo/µRwT and δ/Rw, we can now have therelationship between Fd/ξµRwT as a function of Γo/µRwT for different contactconditions τw/µRw and strain stiffening α . Here τw/µξ = 0 represents frictiondominated condition (τw ≪ µξ ) and µξ/τw = 0 represents adhesion dominated orwear dominated condition.37Figure 3.6: Log-log plot of dimensionless dissipative force at steady-stateFd/ξµRwT vs dimensionless cleavage toughness Γo/µRw at differentvalues of τw/µξ for α = 2, α = 5, and α = 9 using ABAQUSThe correlations between the dimensionless dissipative force Fd/ξµRwT , thedimensionless toughness Γo/µRw, the friction coefficient ξ and the dimensionlessshear resistance or strength due to adhesion and wear τw/µ are obtained from finiteelement analysis (FEA) and reported in Figure 3.6.383.1.1 ResultsFriction Dominated regimeThe numerical results in Figure 3.6 can be fitted to the relationFdµRwT≈ ξ(ΓoµRw)n(3.13)in the extreme regime of high toughness, where Γo ≫ µRw (right extreme in Figure3.6). Note, alternatively, this can also be interpreted as low stiffness and smallradius regime. As discussed previously, this is the condition where the cuttingconfiguration is representative of true-cutting with δ ≈ Rw inFor this case, the power-law coefficient n in Eq. (3.13) is reported in Table3.1 as a function of the strain-stiffening coefficient α . In this regime, Fd is foundto be independent of adhesion and wear due to τw, and mainly dependent on thestrain-stiffening coefficient α (solid and dashed lines in Figure 3.6). Thus, theinterfacial stresses in Eq. (3.9) are dominated by friction, given the high contactpressure P due to the proximity of the wire with the crack tip (δ ≈ Rw). The hightoughness as a friction-dominated regime was also observed in the puncture of softsolids [45]. From Table 3.1 we can deduce that high strain-stiffening α has a lowervalue of the exponent n. In agreement with the observations made in Section 2.3.1,we note that increasing strain-stiffening tendency α , decreases the critical pressureP sustained near contact region. This is because, at high α , the strain-stiffeningmaterial sustains less crack tip deformation at steady-state for a given Γo/µRw.Table 3.1: Power-law coefficient n of the fitting function in Eq. (3.13), in thehigh toughness regime (Γo ≫ µRw) obtained to the least square feet withan r-square accuracy of r2 ≥ 0.98α n2 1.185 1.069 0.96However, by taking n≈ 1 in Eq. (3.13), we have Fd ≈ ξ Γo T , where the dissi-39pative force is nearly independent of the wire radius Rw. Thus,FssT|Γo≫µRw ≈ (1+ξ )Γo (3.14)Adhesion and wear dominated regimeIn the other extreme regime of low toughness in Figure 3.6 (left extreme), Γo ≤µRw, and the dissipative force Fd , from Figure 3.6, can be fitted to the relationindependent of Γo/µRwFdµRwT≈ 4(ξ +τwµ)(3.15)In this regime, as deduced from Eq. (3.15), we have a linear scaling betweenthe dissipative force Fd and the wire radius Rw. This can also be interpreted as highstiffness and/or large radius regime. Unlike the case in high toughness regime,here, we also have a negligible influence of strain stiffening, as shown in Figure 3.6.Moreover, the interfacial forces in Eq. (3.9) are now dominated by the adhesionand/or wear of the specimen τw, given the low contact pressure P in this case. Forthis case, we have the steady-state force Fss asFssT|Γo≤µRw ≈ Γo +4(τw +ξµ)Rw (3.16)describing the low toughness regime.40Cutting Force PredictionTo understand the competition between different dissipative forces on cutting, weprovide a generic expression using Eq. (3.14) and (3.15), we deduce the empiricalrelationFssT≈ (1+ξ )Γo +4(ξµ+ τw)Rw (3.17)which represents all scenarios with a good approximation (goodness of fitshown in Figure 3.6), for the two cases analyzed above. Note that, Eq. (3.17)simplifies to (3.14) or (3.16), in the extreme regimes of high toughness (frictiondominated), and low toughness (adhesion or wear dominated), respectively.Now, for the condition τw ≫ µξ , i.e. when friction coefficient ξ is negligiblysmall, Eq. (3.17) becomesFssT≈ Γo +4τwRw (3.18)To understand the relationship between τw and cleavage toughness Γo, we define alength scale lw = Γo/τw. Now, Eq. (3.18) becomesFssΓoT≈ 1+4Rwlw(3.19)Now, when τw represents adhesion strength at contact, we have τw ∼Wa whereWa in J/m3 is the work of adhesion. From Eq. (3.18), when we have Γo ∼Wf l f ≫WaRw then contact dissipation is ignored and the forces contributing to the scissionof chains at the crack tip are dominant. On the contrary, the energetic cost ofovercoming interfacial shear strength cannot be ignored suggesting a linear scalingfor Fss/T with respect to Rw.Now, given that wear is created by material damage at the interface, supposewe assume τw ∼Wd ∼Wf , where Wd is the dissipative work, characterizing wearor damage in the material at the contact interface. We note one may also interpretτw as the onset of plastic yield at contact interface [4]. In the context of remotetearing tests, Liu et al. note that Wd scales the dissipative fracture energy Γd ∼Wd l f[46] responsible for the development of a large fracture process zone (FPZ). Asdiscussed in 1.2.1, in the case of an SN network gel, this characterizes networkimperfections creating weaker spots in the bulk of the material away from the crack41tip. We saw that, in an ideal material, Γ ≈ Γo since Γd ≈ 0, giving FPZ or l f inO(nm).In cutting, in the presence of material damage in the bulk, we have lw ∼ l f ,with l f = Γo/Wf , the fracto-cohesive length, which is typically in the range of 0.1mm for brittle gels (10 times smaller than le). Based on this assumption, from Eq.(3.19), we note that there exists a transition radius R∗w ≈ l f /4 which defines theforce-radius scaling such that when Rw ≥ R∗w, we can observe the linear correlationbetween Fss/T and Rw. This region marks the scaling (enhancing) of the damagezone by damaging or yielding the material away from the tip at steady-state cutpropagation. In the absence of damage, we have FPZ and cutting localized to thecrack tip akin to the picture presented by Lake and Thomas [8].3.1.2 Cutting ExperimentTo validate the scaling relations in the previous section, we performed simple wire-cutting experiments on PAAm hydrogel (see A.1.1 for sample preparation). Forthis, we measured the force per unit thickness F/T as a function of wire positionD, as shown in Figure 3.7 (a). In particular, we focus on the steady-state cuttingforce Fss/T occurring after the initial force peak required to initiate the cut [47], ashighlighted in Figure 3.7 (a).Now, Figure 3.7 (b) reports the steady-state cutting forces Fss/T for five cut-ting wire radius Rw = 0.125, 0.210, 0.275, 0.390 and 0.470 mm. Here, we reportthe mean of five trails for Fss/T , where the square indicates the median value andthe error bar indicates the standard deviation for data points corresponding to eachwire radius Rw. The tested material behaves like a neo-Hookean hyperelastic solid(A.1.1) and the cutting speed is D˙ = 0.2mm/s.In Figure 3.7 (b), we can observe a linear correlation between Fss and Rw. Thisscaling was predicted in an adhesion or wear-dominated regime for a material withlow Γo/µRw when ξµ ≪ τw in Eq. (3.18). The physical mechanism behind thisphenomenon is discussed in Section 3.2.As reported in A.1.1, the measured toughness from the pure shear test is Γ =2.6 J/m2 and stiffness µ = 685Pa. Now, the elasto-cohesive length of the ma-42Figure 3.7: (a) Force per unit sample thickness (F/T ) vs Wire displacement(D) showing steady-state force per unit thickness (Fss/T ) (b) Relation-ship between Fss/T (mean of five cutting tests) vs wire radius Rw for aPAAm hydrogel, comparing experiments (blue squares) with a linear fit(red solid line), and with indicated Γ= 2.6 J/m2 (black dashed horizon-tal line).terial, based on the reported material properties is le = 3.8 mm, and the fracto-cohesive length is l f = 371µm, giving a transition radius for adhesive/wear sensi-tivity R∗ ≈ l f /4 = 92µm as noted previously and shown in Figure 3.8.The parameter R∗ (see Figure 3.8) is smaller than the smallest wire radius uti-lized in our experiments. Giving us linear dependence of steady-state forces on Rwvia sensitivity to dissipation in the form of adhesion and wear. In Fig. 3.7, the slopeof the line (red dashed) connecting the experimental data points is 31 kPa, which,according to Eq. (3.14), neglecting friction, should be 4τw, giving τw = 7.75 kPa,close to Wf characterizing FPZ.43Figure 3.8: Relationship between Fss/T vs wire radius Rw highlightingcleavage toughness Γo (in a yellow cross) and R∗ (in a blue cross) whichmarks the transition to contact dissipation dominant effect making it lessfavorable than tearing limit estimated from pure shear3.2 DiscussionIn Eq. (3.18), when Rw ≈ 0, we have Fss/T = Γo. Now, in Figure 3.7 (b), the inter-cept between the line connecting the data points and the y-axis gives an estimatedΓo = 1.6 J/m2 (see Figure 3.8 yellow cross), which is smaller than the measuredΓ = 2.6 via pure-shear tests (see Figure 3.8 dashed line in black), as reported inA.1.1. Further, Figure 3.7 (b) reports Γ measured from remote fracture tests suchas pure shear as the horizontal (black dashed) line. We attribute the discrepancybetween the toughness measured via pure shear tests (Γ) and the intercept (Γo) ofthe force-radius plot in Figure 3.7 (b) and 3.8 to the additive contributions towardthe measured toughness Γ= Γo +Γd , where Γo is the cleavage toughness (adopted44in our model), and Γd is the additional contribution due to the formation of the FPZ[48].Consistently, using Y-shaped cutting experiments, Zhang et al. [33] observedthat the cutting force can be smaller than the tearing force or the toughness mea-sured from pure-shear tests. Thus, this theoretical framework and experimentalresults, provide more insight into the physical mechanisms differentiating cuttingfrom tearing or fracture. In cutting, the crack propagation is driven by contact load-ing near the crack tip, and dissipative mechanisms (friction, adhesion, and wear)are controlled by blade sharpness. Whereas, tests such as pure shear, give a macro-scopic measure of toughness, and dissipative contributions (from the FPZ) thatonly depend on the material or sample.Moreover, the linear correlation between the cutting force Fss and the wire ra-dius Rw, by neglecting friction, gives the simple Fss/T = Γo+4τwRw. By assumingthat Fss/T ≈ Γ, where Γ ≈ Γ0 + 4τwRw, we can observe a similarity with the ex-periments performed by Thomas et al. [28], where uniaxial tensile tests of notchedsamples, created by razors of various tip radii R, gave a toughness scaling Γ∼Wf R.In this analogy, the dissipative mechanisms controlled by blunt razor blades havelikely enhanced or scaled FPZ size, thus increasing the apparent toughness of thesamples. This creates cutting resistance as a function of wire/blade geometry.Additionally, we note that wire cutting is highly affected by the three dimen-sionality of the sample, especially at cut initiation as shown in [47]. However, inour steady-state cutting analysis, we assume plane-strain conditions. This is rea-sonable owing to the low toughness of the sample and the three-dimensionalityof the sample is likely to play a marginal role during steady-state cutting. Fur-ther, this is supported by the close validation between theory and experiments. Forthe reported experiments, we have a strong adhesion and wear dominance, givingτw/µ = 11.31≫ ξ , thus justifying the approximation in Eq. (3.19). This impliesthat the elasto-cohesive length must be larger than the fracto-cohesive one, specif-ically, le ≫ ξ lw. Given that in the tested materials le/lw ≈ 10, this means ξ ≪ 10which satisfies the hypothesis if we assume ξ < 1.Moreover, another condition is that of τw ≪ ξµ , for which, Equation (3.16)would also provide a linear scaling between radius and force, but due to frictionand stiffness instead of wear, giving Fss/T ≈ Γo + 4ξµRw. This would require45lw/le ≫ 1/ξ . Thus, by assuming τw is due to wear/damage, for which lw ∼ l f , thematerial, in this case, should have a much larger FPZ l f than the elasto-cohesiveone le i.e. Wf < µ . As discussed by Long et al., [5], this is only the case in verystiff materials and not in soft elastomers and gels which typically have Wf > µ (asnoted in Section 2.3).46Chapter 4Role of Compressibility inCuttingIn this chapter, we extend the ideas presented in the previous chapter for strain-stiffening incompressible materials (K/µ → ∞) to compressible strain-stiffeningmaterials.Now, to understand the role of compressibility in cutting, we consider a com-pressible Ogden hyperelastic strain energy density (W ) which is additively decom-posed as deviatoric strain energy density (WD) and volumetric strain energy density(WV ), and defined asW =WD +WV (4.1)W =2µα2(λ¯α1 + λ¯α2 + λ¯α3 )+K(Jv−1)2 (4.2)where µ (in Pa) is the shear modulus, K (in Pa) is the bulk modulus, and λ¯i =λiJ−1/3v are the deviatoric principal stretches, with λi the corresponding principalstretch and Jv = dv/dv0 the volumetric swelling ratio, where dv and dv0 are theunit volumes in current and reference configuration, respectively.47Here, the dimensionless principal Cauchy stress in direction X1, σ1/µ , is givenbyσ1µ=23αJv(2λ¯α1 − λ¯α2 − λ¯α3 )−2Kµ(Jv−1) (4.3)where the deviatoric component of stress σ1d/µ isσ1dµ=23αJv(2λ¯α1 − λ¯α2 − λ¯α3 ) (4.4)and the volumetric component p/µ ispµ=−2Kµ(Jv−1) (4.5)where p = (σ1 +σ2 +σ3)/3 is the mean stress penalizing volume change. Notethat p/µ (in Eq (4.5) scales linearly with the compressibility factor (K/µ) andJacobian (Jv).4.1 ResultsTo understand the role of compressibility, we follow the procedure adopted pre-viously and obtain the dimensionless energy release rate J/µRw as a function ofdimensionless distance from the center of the wire to the crack tip (δ/Rw) shownin Figure 3.2. Now, Figure 4.1 (a) reports J/µRw vs δ/Rw for incompressible ma-terial (K/µ→∞) (in blue) and a highly compressible material (K/µ = 1) (in red).In particular, when δ ≫ Rw, i.e., when the wire is far away from the crack tip,we note that, J/µRw vs δ/Rw curves for both incompressible (blue) and compress-ible material (red) are independent of strain-stiffening tendency (α) in the material.Further, the results from FEA, give the relationship between them asJµRw= β(Rwδ)2(4.6)where β is the dimensionless proportionality constant.48121211Figure 4.1: (a) Semi-log plot of dimensionless energy release rate J/µRwvs δ/Rw for compressibility K/µ → ∞ (blue) and K/µ = 1 (red) withstrain stiffening parameters α = 2, α = 5, and α = 9 (b) Log-log plot ofdimensionless Cauchy stress σ22/µ vs dimensionless distance from thecrack tip r/Rw at δ = Rw for K/µ → ∞ (blue) and K/µ = 1 (red) withstrain stiffening parameters α = 2, α = 5, and α = 949we note that, while the scaling law with respect to δ/Rw in Eq. (4.6) holds forJ ≫ µRw, irrespective of α and K/µ , the value of the constant β depends on thedegree of compressibility (K/µ).For the two cases shown in Figure 4.1 (a), β is nearly two times higher for theincompressible case (K/µ → ∞) compared to the compressible case (K/µ = 1).Therefore, from these conditions, we can see that for a given J/µRw, the distancefrom the crack tip at steady-state δ/Rw is lower when K/µ is lower (compressible).In other words, at a given distance away from the crack tip, δ/Rw, the energy re-lease rate is lower when the material is compressible. This is because, when K/µis lower, it favors localized volume change or ”sinking” near contact. However,high K/µ causes the energy to flow to the crack tip effectively. I.e., at a high K/µ ,a heavier penalty on the volumetric strain energy allows for a cut to be propagatedat a larger δ/Rw by the wire. Moreover, as noted previously, when δ/Rw = 1 orJ ≫ µRw, we have an indentation-cutting regime [30, 32].Now, to understand the asymptotic stress state at the crack tip, in Figure 4.1(b), we plot the dimensionless stress σ22/µ vs dimensionless distance from thecrack tip r/Rw in the indentation-cutting regime (δ = Rw). For an incompressiblematerial, when δ = Rw, we have σ22/µ ∼ 1/r akin to NLEFM (Nonlinear ElasticFracture Mechanics) highlighting stress uniaxiality (Figure 4.1 (b), blue).Therefore, we have a larger J/µRw at a given δ/Rw. However, in the case ofa compressible material (Figure 4.1 (b), red), we obtain a scaling σ22/µ ∼ 1/√rakin to LEFM highlighting stress triaxiality or the dominance of mean stress penal-izing uniaxial/deviatoric crack tip deformation in X2 independent of α and attainssmaller J/µRw for the same δ/Rw.Figure 4.2 (a), shows the dimensionless Cauchy stress σ22/µ at the crack tip(3.2) as a function of dimensionless distance from the crack tip δ/Rw. As expected,tougher material experiences higher stress in X2 direction at steady-state and it de-creases as a function of δ/Rw or Γo/µRw (J = Γo is adopted). As noted earlierby examining the asymptotic stress fields, we understand the degree of stress uni-axiality at the crack tip by plotting the ratio of stresses σ22 and σ11 as a functionof δ/Rw. Consistent with the results obtained in the pure shear fracture test forsoft materials [41], in cutting, we see that soft-tough materials Γo ≫ µ , sustain auniaxial state of stress with the condition σ22 ≫ σ11 and σ22 ≤ σ11 in the limit of50Figure 4.2: Semi-log plot of (a) dimensionless Cauchy stress σ22/µ vs δ/Rw(b) dimensionless stress ratio σ22/σ11 vs δ/Rw51soft and stiff-brittle materials (Γo/µ).Now, for a stiff-brittle material or a larger wire radius (Γo ≪ µRw), we haveδ ≫ Rw, in this condition, the volumetric component (p/µ) dominates over the de-viatoric component (σ2d/µ) thereby facilitating crack tip propagation over cracktip deformation or blunting (Figure 4.3 (a),(b)). I.e., σ2d/µ ≪ p/µ when δ ≫ Rwor Γo ≪ µRw. On the contrary, for a soft-tough material or a smaller wire radius(Γo ≥ µRw), we have σ2d/µ > p/µ when δ = Rw (Figure 4.3 (a), (b)) highlightingvolume preserving or deviatoric or shear dominant deformations.52Figure 4.3: (a) Semi-log plot of dimensionless deviatoric component ofCauchy stress σ2d/µ vs δ/Rw (d) dimensionless volumetric componentof Cauchy stress p/µ vs δ/Rw for compressibility K/µ→∞ (blue) andK/µ = 1 (red) with strain stiffening parameters α = 2, α = 5, and α = 9534.2 DiscussionWe know that the steady-state force is Fss = ΓoT +Fd . Now, the dissipative forceFd depends on interfacial contact interactions between the wire and the specimensuch as friction, adhesion, and wear. As seen earlier in (4.7), we haveFd = 2Rw T∫ θ2θ1τc sinθdθ (4.7)withτc = ξP+ τw (4.8)as shown in Figure 3.4(a) and Figure 3.4(b).Consistent with the observations made in cutting in-compressible materialspreviously, and in puncture of soft materials [45], we have a friction dominantor low radius regime for a soft-tough material (Γo ≫ µRw), because of high con-tact pressure (ξP ≫ τw in Eq. (4.8)) sustained at steady-state, and adhesive orwear-dominant dissipation in large radius regime for a soft-brittle material (Γo ≤µRw) due to the dominance of shear damage or adhesive dissipative mechanisms(τw ≫ ξP) at contact. This regime can be identified as the scaling of FPZ (FailureProcess Zone) in wire cutting where the scaling of dissipative forces with respectto the wire radius was identified to be independent of strain-stiffening (α).Using FEA, we identify the relationship between Fd ,Γo, µ , ξ , τw and Rw forcompressible materials with different strain-stiffening. Figure 4.4 (a) plots the re-lationship between dimensionless dissipative force Fd/ξµRwT vs dimensionlesstoughness of the material Γo/µRwT for the case τw/µξ = 0.Friction dominated regimeIn the regime of high toughness or friction-dominated regime, the numerical resultsin Figure 4.4 can be fitted to the relation54FdµRwT≈ ξ(ΓoµRw)n(4.9)In this regime, dissipative forces Fd due to friction and the exponent n dependon both α and K/µ . Table 4.1 reports the value of n for two different values ofα . For clarity, only the fitted curves at α = 9 for four different values of K/µ areplotted in Figure 4.4 (b).Table 4.1: Power-law coefficient n of the fitting function in Eq. (4.9), in thehigh toughness regime (Γo ≫ µRw) for α = 5, α = 9 obtained to the leastsquare feet with an r-square accuracy of r2 ≥ 0.98K/µ n (α = 5) n (α = 9)30 1.24 1.192 1.37 1.301 1.41 1.35From this, we can infer that soft-tough materials (Γo ≫ µRw) with high in-compressibility are likely less sensitive to frictional forces (Figure 4.4 (b)) due todecreasing slope with K/µ . This is because of low contact pressure sustained atthe tip at steady-state at a given Γo/µRw in an in-compressible material.55Figure 4.4: (a) Dimensionless dissipative force Fd/ξµRwT vs dimension-less fracture toughness Γo/µRw (b) Dimensionless dissipative forceFd/ξµRwT vs dimensionless fracture toughness Γo/µRw for α = 9 fit-ted and extrapolated with R2 ≥ 0.9756Adhesion and wear dominated regimeIn the adhesive/wear dominant regime (Γo ≤ µRw), we identify the following rela-tionship where m is a dimensionless constant or scaling coefficient.FdξµRwT≈ m(1+τwµξ)(4.10)From Figure 4.4 (a), we see that this linear scaling holds irrespective of bothstrain-stiffening (α) and compressibility (K/µ) of the material in the adhesive orwear dominant regime. However, although the constant (m) is independent ofstrain-stiffening (α), it depends on compressibility K/µ . Now, the steady-stateforce per unit thickness can be estimated to beFssT= Γo +m(ξµ+ τw)Rw (4.11)we see a linear scaling of steady-state forces per unit thickness (Fss/T ) withrespect to wire radius (Rw). Now, when τw ≪ ξµ , which is typically possible invery stiff solids, we then have sensitivity to wire-radius due to friction. Whereas,when τw ≫ ξµ , we have a scaling of forces correlated to adhesion and wear.Figure 4.5: Co-efficient m depicting shear sensitivity as a function of com-pressibility ratio µ/K57Now, Eq. (4.11) reduces toFssT= Γo +mτwRw (4.12)Eq. (4.12) was fitted with an R2 ≥ 0.98, to identify the constant (m) and we plotthe values as a function of µ/K ranging from 0 to 1 where µ/K = 0 representsincompressible limit and µ/K = 1 represents very high compressibility. In Figure4.5, we see that, m decreases drastically with an increase in compressibility (K/µ)of the material. I.e., we see that, at a given Γo/µRw, soft-brittle materials (Γo ≪µRw) with high incompressibility are more sensitive to dissipative mechanisms inthe form of wear or adhesion.58Chapter 5ConclusionOver the past few decades, we have gained mechanistic insight by examining thefailure of soft materials subject to triaxial expansion (cavitation) tests and remoteloadings (pure shear, split tear, and tension test). However, only a few studies focuson contact-mediated crack propagation via wire cutting. By looking at the stressfield around the crack tip in pure shear conditions, we note 1/r stress singularityirrespective of α . However, the elasto-cohesive lengthscale Γ/µ does not accountfor α . Therefore, the stress state and critical stretch at failure are different de-pending on strain-stiffening. The consequences of constitutive behavior such asstrain-stiffening α and compressibility K/µ were then studied in the context ofwire cutting.In this thesis, through finite element analysis, dimensional arguments, andwire-cutting experiments, we note the following:• When Γo ≪ µRw, we have far-field cutting regime valid for soft-tough orlow cutting tool radius.• When Γo ≫ µRw, we have indentation cutting regime valid for stiff-brittleor large cutting tool radius.Moreover, in the absence of dissipation, Fss = ΓoT is independent of tool geometryand the constitutive behavior of the material.59Now, by incorporating dissipative mechanisms using the modified Coloumb’slaw τc = ξP+ τw, we see the following:• Tough materials are sensitive to friction owing to deep indentation or highcontact pressure (P) at steady-state (independent of τw)• Brittle materials are vulnerable to adhesion and wear (independent of ξ dueto low P).By systematically varying the strain-stiffening tendency (α) of an in-compressiblehyperelastic material, we note that• Adhesion or wear dominant regimes (in the limit of brittle materials) areindependent of α and the force Fd is linear in Rw as Fd ≈ 4τwRwT• Frictional dissipation depends on α (in the limit of tough materials). Thedependence of Fd on Rw (although weak and not linear as noted previously)is larger for materials with low α . Higher strain-stiffening decreases thesensitivity of cutting forces to tool geometry.Further, we identify R∗ as the transition radius which scales the transition fromradius-independence to a regime of linear dependence of Fss with Rw. This is con-sistent with the observation made by [32, 33] although noted for thin films with along crack with a notch angle of 32o. We identify the force scaling due to sheardamage akin to scaling of the apparent toughness or FPZ size due to dissipativemechanisms not confined to the crack tip.Finally, by varying the compressibility K/µ from the limit of in-compressibleK/µ → ∞, to an extreme case of very high compressibility K/µ = 1, we notethat• Adhesion or wear dominant regimes (in the limit of brittle materials) areindependent of α and the force Fd is linear in Rw. However, the scalingcoefficient m (in Fd = mτwRwT ) is a function of compressibility. With mlargest for an incompressible material.60• Frictional dissipation depends on α (in the limit of tough materials) but alsoon compressibility K/µ . For the case with α = 9, the dependence of Fd onRw is larger for materials with low K/µ . Higher compressibility increasesthe sensitivity of cutting forces to tool geometry in this regime. This holdsirrespective of α .The proposed model provides a valuable tool to predict Γo from the interceptat Rw = 0 and correlates the dissipative process zone scaling with interfacial shearstrength due to adhesion or wear τw from the slope of Fss vs Rw in wire-cutting.This suggests that, in the regime of wear/adhesion dominance, sharp wires pro-duce clean cuts and blunt wires produce rougher cut surfaces requiring force perunit thickness to be larger than cleavage toughness Γo.We also note that these conclusions are restricted to two-dimensional steady-state cutting. For the case where the cut is mediated by a combination of in-planeand out-of-plane forces, we may have non-uniform stresses along the thickness di-rection (in X3 direction) causing failure to initiate at one of the edges (free-ends) asnoted in [47]. Reyssat et al. [49], studied the cutting initiation phase as mediatedby frictional shear stress, cutting soft agar gel samples. The role of out-of-planeshearing in steady-state wire cutting was not quantitatively explored. Intuitively,we can deduce a benefit in the combination of indentation and shearing as mediatedby the out-of-plane motion of the wire. From the observations made in this study,one can deduce that slicing is beneficial only in the presence of significant fric-tional forces between the wire and the sample. This is the case in which we havehigh contact pressure, high frictional coefficient, high adhesive shear strength, or acombination of them.5.1 Future WorkNature is optimal and produces soft, viscous, ordered, and directional materials.Performing wire-cutting experiments on materials with a rate-dependent hystere-sis and/or anisotropic constitutive behavior would be an interesting future direc-tion. This is of practical importance because several biomaterials such as skin,heart valves [50], and brain [16], have strong preferential deformation modes. Thiscan be explored using the UANISOHY PER INV subroutine in the commercial61software package ABAQUS. This would allow us to incorporate a hyperelasticanisotropic response with fiber directions by splitting strain energy density intoisotropic and anisotropic contributions ex: Gasser-Ogden-Hozpfel model [51] forsingle and two fiber family. The role of anisotropy in soft materials in the cuttingand fracture is an exciting new direction.Further experimentation and research must be undertaken to establish univer-sally accepted failure criteria for soft solids (stress or energy-based). Moreover, thecorrelation between wear/shear damage (due to contact shear stresses) and the workof fracture determined from tensile tests for soft materials have to be carefully stud-ied for different classes of materials. This would require exhaustive imaging andmaterial characterization to capture different failure mechanisms and associatedtheoretical explanations at continuum scales useful for engineering applications.Additionally, we note that the conclusions drawn on the thesis are based onthe well-known Coulomb’s law or dry friction model. However, soft materialssuch as hydrogels are multi-phasic. They may self-lubricate when subject to largecontact forces and can have contact conditions where the contact surfaces can becompletely separated by fluid lubricant of considerable thickness (typically largerthan surface asperities). Under these conditions, we can have a condition in whichthe contacting solids are fully separated by a film of lubricant to which the contactloads due to indentation are transferred by the hydrodynamic pressure developed inthe fluid. This is called the elastohydrodynamic lubrication (EHL) regime [40, 52].The case where there is partial solid-solid contact is called a mixed lubrication(ML) regime [40]. Using a sliding friction experiment setup, Dong et al., study theeffect of frictional slicing and failure initiation in soft materials by accounting fortransitions from EHL to ML regimes at the leading contact edges (front face) [40].Some other works that focus on these frictional instabilities and failure in soft solidcontact include the works in [38, 39]. 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PhD thesis,University of Colorado at Boulder, 2019. → page 7170Appendix ASupporting MaterialsA.1 ExperimentsMechanical properties of soft materials such as elastomers and gels are often vis-coelastic and poroelastic [64]. Experimentally, we can minimize these rate-dependentdissipative effects by taking measurements from tests carried out at a low stretchrate. For example, in an elastomeric gel, viscoelasticity results from sliding be-tween the polymer chains causing viscoelastic hysteresis [64]. Now, the time ofviscoelastic relaxation, τv is independent of the length characteristic of the sampleor other dimensions and therefore a material property. However, poroelasticity re-sults from the migration of the solvent (in multiphasic materials), where the timeof poroelastic relaxation depends on the sample geometry and boundary conditions[65].Viscoelastic relaxation time τv and let D be the diffusivity of the solvent in thenetwork. We obtain a material-specific length scale√Dτv which represents thedistance of the spread of the solvent within a time comparable to the time of vis-coelastic relaxation [64]. Thus τv represents viscoelastic relaxation and t ∼ L2/Drepresents time and length of poroelastic relaxation where L is the length of thesample [64, 65]. For Polyacrylamide (PAAm) sample, viscoelastic relaxation timeis in the order of 1s [64] and the diffusivity of the water molecule in PAAm ison the order of D ∼ 10−10 m2/s [63]. Thus, the time needed for poroelasticity toprevail t ∼ L2/D ∼ 105s. Now, When the experimental time scale (tex) is of the71order where t ≪ tex ≪ L2/D, viscoelastic relaxation has completed and poroelasticrelaxation is yet to start. This suggests that viscoelasticity is important at shortertime scales and poroelasticity is important at longer time scales [64]. In our ex-periments, care was taken to make sure that tex falls within this range so that rateeffects can be neglected [64].72A.1.1 Material CharacterizationSynthesis of polyacrylamide (PAAm) gel: We purchased acrylamide AAm, A8887,N,N’ Methylenebisacrylamide (MBAA, M7279), tetramethylethylenediamine, andammonium persulfate (APS) from Sigma Aldrich. The polyacrylamide (PAAm)hydrogel was prepared by mixing 10 mL of acrylamide solution (40% w/w), 1.5mL of MBAA (2% w/w), 15 µL of TEMED, and 1 mL of APS (66 mg/mL). Themixture was then injected into a closed acrylic mold overnight.Fracture toughness can be measured from the Pure Shear test by taking anunnotched sample and stretching it along its height (H) to its ultimate stretch (λ f )(Figure 2.5 (a)). This is followed by introducing a crack of length (c) and stretchingthe notched sample until the sample fails catastrophically (Figure 2.5 (a)). Themaximum stretch of the notched sample is denoted by critical stretch, λc. Now, thetotal energy stored in the sample at the critical stretch is U =W (λc)H.L.T where H,L, and T are the undeformed height, length, and thickness of the sample. W (λc) isthe strain energy density, calculated as the area under the engineering stress (σeng)vs stretch (λ ) of the un-notched sample at λc (shaded in sky blue color, Figure A.1(b)). Toughness measured using the pure shear test is given byΓ=W (λc)H (A.1)Figure A.1 (a) Shows the INSTRON setup for a notched pure shear samplepulled vertically. Figure A.1 (c) shows the pure shear response of a tough PAAmgel with Γ = 2.6J/m2. We then examined a uniaxial tensile specimen (dog-bonesample) to characterize an as-prepared PAAm material and was fitted to the neo-Hookean eng stress-stretch relationship (α = 2). From the initial slope of the stress-stretch plot, we obtained µ ≈ 685 Pa and area under the curve, work of fractureWf ≈ 7100 J/m3 (mean of three samples).73Figure A.1: (a) Uniaxial Tension response of soft material σeng vs Stretch λ(b) Pure Shear INSTRON machine (c) Pure Shear response of a toughsoft material σeng vs Stretch λ for an Unnotched sample (red) and aNotched sample (blue) (d) Pure Shear response of a brittle soft materialσeng vs Stretch λ for an Unnotched sample (red) and a Notched sample(blue)74
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Bacca, Mattia
Cutting mechanics of soft solids
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