Modeling of competition between shear yielding and crazing in amorphous polymers’ scratch

Accepted Manuscript

Modeling of competition between shear yielding and crazing in amorphous polymers’ scratch Han Jiang , Jianwei Zhang , Zhuoran Yang , Chengkai Jiang , Guozheng Kang PII: DOI: Reference:

S0020-7683(17)30305-0 10.1016/j.ijsolstr.2017.06.033 SAS 9640

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

21 October 2016 27 May 2017 26 June 2017

Please cite this article as: Han Jiang , Jianwei Zhang , Zhuoran Yang , Chengkai Jiang , Guozheng Kang , Modeling of competition between shear yielding and crazing in amorphous polymers’ scratch, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.06.033

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Highlights  A crazing initiation criterion was proposed to constitutively considering competition between shear and craze;  Scratch damage mechanisms of ductile PC and brittle PMMA were investigated;  Shear yielding dominates the scratch process of PC;  Coexistence and competition between shear and craze are main mechanisms for

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PMMA.

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Modeling of competition between shear yielding and crazing in amorphous polymers’ scratch Han Jianga,*, Jianwei Zhanga,b, Zhuoran Yanga, Chengkai Jianga, Guozheng Kanga Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province,

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a

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China b

School of Mechanics and Engineering Science, Zhengzhou University, Zhengzhou,

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Henan 450001, China

*Corresponding author. Tel: +86-28-87601442; fax: +86-28-87600797

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E-mail address: [email protected]

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Abstract: An effective approach to investigate the complex scratch damage mechanisms of polymers, based on a suitable material constitutive model, is important. For the constitutive model capable of the description of the competition between shear yielding and crazing of amorphous polymers, a crazing initiation criterion was proposed. The scratch damage mechanisms of ductile polycarbonate (PC) and brittle poly (methylmethacrylate)

(PMMA)

were

experimentally

and

numerically

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investigated. It can be found that, the shear yielding dominates the scratch process of PC, while the coexistence and competition between shear yielding and crazing are the main damage mechanisms for PMMA scratch. The effect of scratch velocity was also investigated. Those findings are helpful for the designing of scratch-resistant

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polymers.

Key words: Polymer, constitutive model, scratch, damage mechanism, finite element simulation 1. Introduction

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The surface scratch performance is important for the aesthetics, structural integrity, and durability of polymer components (Briscoe et al., 1996; Gauthier and Schirrer,

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2000; Brostow et al., 2010; Browning et al., 2013, Barr et al., 2016). Different polymeric materials show various scratch deformation patterns (Xiang et al., 2001;

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Jiang et al., 2009). Generally speaking, two types of damage have been commonly observed for the scratch of amorphous polymers: the ductile type of damage (related to

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shear yielding) and brittle damage (related to crazing) (Xiang et al., 2001, Jiang et al., 2009, Barr et al., 2016). In their experimental observation (Xiang et al., 2001), while

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both shear yielding and crazing damage mechanisms were confirmed in the scratch of brittle polystyrene, only shear yielding was found in the ductile polycarbonate. Using the effective plastic strain and volumetric strain as the nominal indicators for shear yielding and crazing respectively, Lim (2005) suggested that crazing could be another mechanism of scratch damage of polymers competing with shear yielding. It is clear that shear yielding and crazing may coexist and compete against each other during the scratch process of amorphous polymers at certain conditions. Due to their inherent 3 / 57

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rate-dependent nature, polymers’ scratch damage behavior are significantly influenced by scratch velocity. A decrease of scratch velocity leads to a delay of scratch damage for thermoplastic olefins (TPO) and polypropylene (PP), respectively (Browning et al., 2008; Chivatanasoontorn et al., 2012). An increase of scratch velocity changes the scratch damage of soft TPO from a ductile mode to a brittle mode (Browning et al., 2008; Jiang et al., 2009).

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For a polymer scratch process which should consider the rate-dependent nonlinear constitutive relationship and complex damage mechanisms, as well as contact behavior, obtaining an analytical solution is nearly impossible. When adopting a suitable constitutive model, the finite element method (FEM) can be a good choice to handle

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this issue. Using the simplified constitutive models such as elastic-perfectly-plastic model and piece-wise linear constitutive model, a series of parameter studies were successfully conducted to investigate the effect of mechanical parameters of polymers on the scratch behavior using commercial FEM software (Jiang et al., 2007; Hossain et

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al., 2011, 2012). However, those oversimplified constitutive models only gave semi-quantitative results, without mentioning their incapability for describing complex

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scratch damage mechanisms. Some works have been attempted to constitutively include the effect of materials’ nonlinearity when describing the polymer scratch

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behavior (Lee et al., 2001; Aleksy et al., 2010; van Breemen et al., 2012a; Krop et al., 2016). Lee et al. (2001) adopted Boyce-Parks-Argon (BPA) model (Boyce et al., 1988)

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to describe the evolution of the stress distribution and the equivalent plastic strain during the scratch process of polycarbonate for a simplified 2-D problem. Aleksy et al.

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(2010) used a viscoelastic-viscoplastic model without considering the strain hardening to investigate the time-dependent recovery for the scratch of PMMA. van Breemen et al. (2012) utilized Eindhoven glassy polymer (EGP) constitutive model to study the friction properties of polycarbonate. Hossain et al. (2014) employed a rate and pressure dependent constitutive model to study the scratch deformation of polycarbonate and styrene-acrylonitrile. While most works focused on the ductile type (shear-yielding) of scratch behavior, neither the brittle type (crazing) of scratch damage nor the possible 4 / 57

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coexistence and competition between shear-yielding and crazing has been discussed. Extensive attention has been paid to the constitutive modelling of amorphous polymers considering shear yielding (Haward and Thackray, 1968; Boyce et al., 1988; Arruda et al., 1993; Wu and Van der Giessen, 1993; Anand and Gurtin, 2003; Mulliken and Boyce, 2006; Richeton et al., 2007; Cao et al., 2014). Crazing, which often results in brittle behavior, has also drawn much attention (Argon, 2011; Kramer, 1983; Kramer

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and Berger, 1990). Estevez et al. (2000) and Socrate et al. (2001) adopted the cohesive surface modeling to model the craze-initiation, widening and breakdown. As noted by Gearing and Anand (2004), not only the orientation of crack nucleation, but also its propagation is numerically restrained when using the cohesive surface modeling.

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Gearing and Anand (2004) proposed a continuum method to model the competition between shear yielding and crazing according to a switch rule of possible plastic flow. It is also well known that strain rate can cause the ductile-brittle transition in amorphous polymers (Haward et al., 1969; Ishikawa et al., 1981). Miehe et al. (2015) utilized a

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continuum phase field model to describe the ductile-brittle transition in amorphous polymers. Linear elastic fracture mechanics has been frequently used for the brittle

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fracture of amorphous polymers (Williams, 1984; Kinloch and Young, 2013). However, this approach considers neither the process of craze initiation, widening and breakdown,

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nor the coexistence of shear yielding and crazing (Estevez et al., 2000; Gearing and Anand, 2004).

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To reveal the complicated scratch damage mechanisms of amorphous polymers (brittle PMMA and ductile PC as two good representatives), a constitutive model which

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can consider the strain rate effect on the competition between shear yielding and crazing is proposed in this paper. After being successfully implemented into ABAQUS with a user material subroutine (VUMAT), the capability of the constitutive model is validated by the tensile test results of thin PMMA plate with a circular hole. Then, with the validated constitutive model, the FEM simulation is conducted to study the complicated damage mechanisms in the scratch process of PMMA and PC. 2. Experiment and finite element modeling 5 / 57

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2.1 Experiment 2.1.1 Materials Commercially available PC 0703 (Sabic, Saudi Arabia) and PMMA V100 (Arkema, France) were utilized in this work. All specimens were heat-treated (90℃ for PMMA and 135℃ for PC, respectively) to minimize the effect of thermo-mechanical

four identical specimens were tested for each condition. 2.1.2 Uniaxial test

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history. All the tests were performed at room temperature. To check the repeatability,

The uniaxial tests of PC and PMMA were conducted on a MTS 858 BIONIX who owns the force capacity of 5kN in axial direction and 20N·m in torsional direction.

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Only the axial loading test was performed in the present work. The specimens for uniaxial tensile tests were planar dumbbell shaped, as shown in Fig. 1, which were also used for the scratch tests. The gauge length was 80mm with a cross-section area of 10mm*4mm. The tensile tests of PC and PMMA were strain-controlled. The strain rate

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of PC is 5e-2/s, 5e-3/s and 5e-4/s, as well as 1e-2/s and 1e-3/s and 1e-4/s for PMMA. For the tensile test of PC, the loading process was stopped when the necking occurs to

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protect the extensometer. For uniaxial compression tests, the specimens were in cylinder shape with diameter of 6mm and height of 6mm. The displacement rates for

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the uniaxial compression tests of PC and PMMA were set as 0.003mm/s, 0.03mm/s and

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0.3mm/s to obtain the strain rate of 5e-4/s, 5e-3/s and 5e-2/s.

Fig. 1. The geometry of the specimen for tensile tests and scratch tests.

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2.1.3 Tensile tests of PMMA thin plate with a circular hole The tensile tests of thin PMMA plate with a circular hole were performed with the

same MTS858 BIONIX. The geometry of the specimen was shown in Fig. 2(a). To investigate its effect, two of loading rates, i.e., 0.005mm/s and 0.5mm/s, were used. A non-contact three-dimensional digital image correlation (DIC) system ARAMIS 5M (GOM mhH Ltd., Germany) was utilized to measure the surface strain field of the specimen during the whole loading process until the total fracture occurred. The facet 6 / 57

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size is set as 19, the facet step is set as 15. The focal length of prime lens is 50mm. The distance between the camera and the specimen is about 0.6m. The obtained results has not been filtered since the local strain of the PMMA plate is large enough.

Fig. 2. (a) The geometry of the specimen of PMMA thin plate with a circular hole, (b) Finite element mesh near the hole.

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2.1.4 Scratch test

The scratch tests of ductile PC and brittle PMMA were performed on a self-developed scratch machine whose detail can be found in the literature (Cheng et al., 2016). The diameter of the stainless-steel scratch tip was 1mm. The specimens of PC

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and PMMA used for scratch tests were the same as those for tensile tests. Two scratch velocities, i.e., 1mm/s and 100mm/s, were chosen to study the loading rate effect on the scratch performance. The scratch distance was 10mm. The scratch test was displacement controlled with scratch depth from 0 to 0.12mm. Due to the limitation of

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the self-developed scratch machine, only the normal force was measured during scratch test. Then, a Canon LiDE 110 photo scanner and a Keyence VHX-100 digital

PMMA substrate.

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microscope were used to investigate the damage features of the scratched PC and

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2.2 Finite element modeling

Since an analytical approach is nearly impossible for the complex scratch behavior

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of amorphous polymers, the FEM simulation with a commercial software ABAQUS (Hibbit et al., 2010) was adopted in this work to study the scratch behaviors of PC and

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PMMA. The constitutive model, as proposed in Section 3 and implemented by a user material subroutine (VUMAT), was used for the FEM simulation. Dynamic explicit step was utilized in the finite element modeling. First, the PMMA thin plate with a circular hole under tension was modeled with a fine mesh as shown in Fig. 2b to validate the constitutive model. The C3D8R (8-node linear brick reduced integration) element was used and the number of elements is 37700. During the tensile process, the bottom surface in the length direction was constrained, 7 / 57

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while the upper surface was loaded at a constant displacement rate of 0.005mm/s or 0.5mm/s. Actual loading period are 600s and 6s for the loading rates of 0.005mm/s and 0. 5mm/s, respectively.

Fig. 3. Schematic plot of finite element model for scratch simulation.

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Then, the finite element model, as shown in Fig. 3, was adopted for the investigation of polymer scratch. Considering the symmetry, a half model was employed. The region along the scratch path was fine meshed to improve the accuracy. The number of elements is 220920 with the minimum element size of 4.17μm*4.17

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μm*4.17μm. Convergence study was conducted to guarantee the numerical accuracy. To determine the coefficient of friction for the contact pairs of stainless steel and PC (or PMMA), the stainless block (63mm*63mm*63mm, the mass of stainless block about 200g) was driven to slide at the surface of PC (PMMA) at a velocity of 100mm/min.

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Coefficients of friction were experimentally determined as 0.36 and 0.26 for PMMA and PC, respectively. The scratch tip was modeled as an analytical rigid body with all of

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its rotational freedoms fixed. Both ends of the polymer panel were fixed. The specimen bottom was restrained in y direction.

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3. Constitutive model and its validation 3.1 Constitutive model

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Referring the theoretical framework of elastic-viscoplastic constitutive model for amorphous polymeric materials (Gearing and Anand, 2004), a crazing initiation

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criterion was proposed to consider the effect of strain rate on the competition between shear yielding and crazing. 3.1.1 Constitutive model considering shear yielding

Fig. 4. A one-dimensional rheological representation of the constitutive model considering shear yielding.

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To illustrate the mechanism of ductile deformation of amorphous polymer, a one-dimensional rheological model for shear yielding is given in Fig. 4, which follows the pioneering work of Haward and Thackray (1968). The stress equation in the relaxed configuration for the linear spring in Fig. 4 is given as Te  2GEe0  K  trEe  1

(1)

T

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It can be transformed to the current configuration as Cauchy stress by

1 e e eT F T F . Ee is the elastic part of Green-St.Venant strain tensor and Ee0 is its J

deviatoric part. The expression of Ee is

Ee 

1 e 1 C  1   F eT F e  1 , where  2 2

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Fe  FF p 1 , F is the deformation gradient and F e and F p are the elastic and

plastic parts of F . G is the shear modulus. K is the bulk modulus. G and K can be obtained by Young’s modulus E and Poison’s ratio υ following the classical theory of elasticity.

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Referring to Richeton et al. (2005), the effect of strain rate on Young’s modulus is

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phenomenally considered here as

  D  E  E0 1  k E *log      D0   

D  De  Fe DpFe1 is deformation rate tensor and

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Here,

(2)

D  D : D is its

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 (Gurtin, magnitude. In the case of small deformation, D is equal to the strain rate ε

2003). E0 is the Young’s modulus at D0 . k E is the strain rate dependent coefficient.

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When kE  0 , Young’s modulus is not sensitive to strain rate. It is assumed that strain rate does not affect Poison’s ratio υ . For shear yielding, Dp  DSp , with DSp is the plastic deformation rate tensor for shear flow. The evolution function of DSp is shown as below (Anand and Gurtin, 2003):

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 T0e  S back  D γ   2τ   p S

here τ 

T0e  S back

p

(3)

1 is the equivalent shear stress. T0e  Te  trTe 1 is the 3

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deviatoric part of Te . S back is the back stress. Adopting the Argon double-kink theory

 As   τ 5 6   γ = γ0 exp 1 -      θ   s    p

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(Argon, 1973), the equivalent plastic shear-strain rate γ p s is: (4)

here γ0 is the pre-exponential shear rate factor. θ is the absolute temperature.

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s  0.077G 1  υ  is the athermal shear strength. The activation volume/Boltzmann’s

constant is proportional to A.

To consider the effect of pressure, Boyce et al. (1988) replaced s by s - ασ m , in

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1 which α is the pressure-sensitivity parameter, σ m  trT is the mean stress. To 3 further consider strain softening, Boyce et al. (1988) assumed that s evolves with

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s equivalent plastic shear-strain rate via s = h(1 - )γ p while s is the saturation value s

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of s . Although already widely used (Arruda et al., 1995; Miehe et al., 2009; Wu and Van der Giessen, 1993), BPA model does not consider the non-linear stress-strain

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response preceding the yield peak. Anand and Gurtin (2003) introduced the evolution of free volume to describe the non-linearity before the yield peak. The evolution of free

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volume η is given as η=g0 (

s -1) γ p scv

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where g 0 is a material constant. The saturation value of s is scv . In this work, the idea of free volume is applied into Argon double kink theory. To consider the coupled relationship between s and η , the s in the evolution function 10 / 57

ACCEPTED MANUSCRIPT of s is rewritten as s=scv 1+b  ηcv -η  . The saturation value of η is ηcv , while its initial value is taken as 0. b is a material constant. The back stress S back is controlled by the nonlinear Langevin spring, which is modeled as the 8-chain model (Arruda and Boyce, 1993) as the following equation

μR λL 3 λp

1

 λp  P   B0  λL 

where μR is the rubbery modulus, λL is the network locking stretch. inverse of Langevin function

 x  = coth  x  -

(6)

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S back 

1

x

is the

1 1 . B0P  B p  trB p 1 is the deviatoric x 3 1 trB p is the effective plastic 3

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part of left plastic Cauchy-Green tensor B p . λp  stretch.

A shear failure criterion λp  λpf , as proposed by Gearing and Anand (2004) and

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verified by Torres et al. (2016), is adopted for the ductile fracture caused by shear flow,

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in which λ pf is the critical effective plastic stretch. 3.1.3 Modification of the constitutive model to incorporate crazing

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Instead of considering materials’ microstructural details, a continuum mechanics approach is adopted to model the craze initiation, widening and breakdown, referring

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to Gearing and Anand (2004). Since craze generally involves with the local micro-voiding, the effect of positive mean stress has been included for the initiation

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criteria (Sternstein and Myers, 1973; Sternstein and Ongchin, 1969). Considering the effects of both mean stress and maximum principle stress, Estevez et al. (2000) and Gearing et al. (2004) thought that craze occurs only when the value of maximum principle stress satisfies the criterion which is a function of the positive mean stress. In this paper, a craze initiation criterion considering the effect of strain rate is proposed as σ1  σ1,cr  c1+

c2 +c3 ln( D )  0 and σ m  0 σm 11 / 57

(7)

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here c1 , c2 and c3 are material constants. σ1 is the maximum principle stress. σ1,cr is the critical value of craze initiation. A switching factor χ is adopted for the transition of material flow from shear to craze (Gearing and Anand, 2004). If the craze initiation criterion (Equation (7)) is

χ  0 , shear flow occurs.  1, χ  0,

Dp  DCp Dp  DSp

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satisfied, χ  1 , which indicates that the craze flow of material begins; otherwise,

(8)

Same as Gearing and Anand (2004), the craze flow rule is phenomenologically

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given as,

DCp  ξ peˆ1  eˆ1 1

(9)

 σ m Here ξ p =ξ0  1  . ξ 0 is the reference strain rate. scraze is the resistance for craze  scraze 

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flow. m is the strain-rate sensitivity parameter. Craze strain, expressed as

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εcraze = ξ p dt , is a good index for the craze flow. When the fracture criterion is satisfied,

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the brittle breakdown of material occurs: f εcraze  εcraze

(10)

3.2 Material parameters identification for PC and PMMA

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As for PC, craze does not occur under our experimental condition. This could be

attributed to reasons that neither the hydrostatic tension is large enough nor the

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loading rate is sufficiently high to initiate craze as required by Equation (7), as the rate-dependence of craze initiation has been observed experimentally by Haward et al (1969). Thus, the constitutive model for PC can be safely simplified assuming no craze initiation. The material parameters for PC without the craze-related parameters, obtained through the procedure described in Appendix A, are listed in Table 1.

Table 1 Material parameters for PC 12 / 57

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Fig. 5. Comparison between experiment data and predicted curves of PC under different strain rates: (a)tension, (b)compression.

Figure 5 shows the comparison between the test data and predicted stress-strain curves of PC. The rate-dependent tensile behavior of PC can be described well in Fig.

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5a. It should be noted that the experimental stress-strain curves in Fig 5a do not show the part of post yield softening as in the compressive condition (Fig 5b) because of the occurrence of necking. The linear section of stress-strain curve, the non-linear behavior before yield peak, strain-softening, strain hardening and the unloading curve

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can be well captured, and well agree with the compression test results, as shown in Fig 5b.

For PMMA, both shear flow and craze flow can be observed under complex stress state. Following the detailed procedure described in Appendix A, the material

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parameters of PMMA are identified, as shown in Table 2.

The experimental and predicted stress-strain curves of PMMA under tension and

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compression at different strain rates are plotted. From Fig. 6a, it can be found that the rate dependence of Young’s modulus and elongation at break under tension is well

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described and coincides with the experimental results. In Fig. 6b, the calculated compressive stress-strain curves are consistent well with the test data. both the brittle

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fracture under tension and ductile deformation under compression have been reasonablly described, using the same set of material parameters as listed in Table 2. It

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should be noted the over-prediction in the post-yield regime may due to the themorheological complex behavior, such as the secondary molecular motions, of PMMA (Mulliken and Boyce, 2006; van Breemen et al., 2012b). Further work is worth to be conducted for considering the inherted complex material behaviors.

Table 2 Material parameters for PMMA

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Fig. 6. Comparison between experiment data and predicted curves of PMMA under different strain rates: (a)tension, (b)compression.

For the strain rate effect on the craze initiation of PMMA, it can be found from Fig. 7 that the stress level at craze initiation increases with the increase of applied

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strain rate. This agrees with the experimental observation of Haward et al. (1969).

Fig. 7. Strain rate effect on the craze initiation of PMMA. 3.3 Validation of the constitutive model

To validate the robustness of constitutive model under complex stress state, the

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comparison between the experimental and the numerical results of tension of the thin PMMA plate with a circular hole at different displacement loading rates is performed.

Fig. 8. Effect of displacement rate on the force-displacement curve of PMMA plate

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with a circular hole under tension.

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As implemented with a VUMAT subroutine, the constitutive mode, is used for the simulation. Fig. 8 shows the load-displacement curves of both experimental and

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numerical results. The rate-sensitivity character of PMMA, as well as the magnitude of displacements at breakage under different loading rates, agree well with the

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experimental results.

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Fig. 9. Comparison between DIC and FEM results of strain distribution along the

tensile direction under 0.5mm/s: (a) DIC result; (b) FEM result. Loading direction is vertical.

Fig. 10. Comparison between DIC and FEM results of strain distribution along the tensile direction under 0.005mm/s: (a) DIC result; (b) FEM result. Loading direction is vertical. 14 / 57

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The strain fields along the tensile direction (Strain Y) before the occurrence of fracture, from DIC observation (Fig. 9a and 10a) and numerical simulation (Fig. 9b and 10b), are shown for the loading rates of 0.5mm/s and 0.005mm/s, respectively. The strain distribution from the DIC results agrees quite well with the FEM results. It can also be found that the maximum Strain Y before break increases significantly with

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the decrease of loading rate.

Fig. 11. Contour plots of εcraze : (a) craze initiates, (b) before crack occurs, (c) crack

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occurs, (d) specimen separates into two parts. Loading direction is vertical.

Meanwhile, the process of craze initiation, widening till breakdown can also be described with the constitutive model. Under a relatively small load, shear flow occurs firstly, since the craze initiation criterion has not been met yet. As the load

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increases, craze initiation occurs at the opposite edges of the hole (Fig. 11a). The magnitudes of craze strain and the size of craze flow region keep increasing with the

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increase of load (Fig. 11b). The fracture criterion (Equation (10)) is met and the craze flow induced crack begins to emerge, as shown Fig. 11c. Finally, a smooth fracture

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surface propagates through the whole specimen, as shown in Fig. 11d. It is clear that the constitutive model can describe not only the competition between shear flow and

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craze flow, but also the effect of loading rate. 4. Scratch damage analysis of ductile PC and brittle PMMA

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4.1 Scratch damage analysis of ductile PC Fig. 12a is the optical scanning image of a scratched PC sample. It can be found

that, with the progressive increase of scratch depth, PC shows different scratch damage patterns, such as whitening caused by minor surface deformation, and material removal induced by severe ductile damage (Fig. 12b), which is similar to the literatures (Jiang et al., 2009; Zhang et al., 2016).

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Fig. 12. Scratch damage phenomenon of PC: (a) optical scanning image of a scratched sample, (b) microscopic images of scratch damage.

From the experimental and numerical results of the normal force-scratch depth shown in Fig. 13, an increase of scratch velocity leads to an increase of scratch normal force for PC. The numerical model can quantitatively predict the velocity

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effect on the normal force-scratch depth curves of ductile PC.

Fig. 13. Comparison between experimental normal force-scratch depth curves and

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FEM results for PC scratch.

Fig. 14. Evolution of effective plastic stretch during PC scratch at different scratch depth: (a) 0.011mm, (b)0.094mm, (c)0.0943mm, (d)0.105mm.

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The evolution of effective plastic stretch, as the index of shear flow, is shown in Fig. 14. Firstly, the effective plastic stretch occurs at the subsurface of PC (Fig. 14a.)

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With the gradual increase of scratch depth, the region enduring severe shear flow expands to the substrate surface (Fig 14b). Finally, the ductile fracture criterion is

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reached. Thus, the material fracture induced by shear flow initiates at the subsurface (Fig. 14c), and then propagates to the surface of PC (Fig. 14d). Xiang et al. (2001)

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also experimentally observed the shear deformation initiated from the subsurface during the PC’s scratch process.

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Fig. 15 shows the FEM results and the image of scratch-induced fracture of PC at

a scratch velocity of 100mm/s. The fracture phenomenon observed in the scratch test (Fig. 15b) is well captured by the FEM simulation.

Fig. 15. Scratch induced fracture of PC: (a) FEM result, (b) corresponding test data. 4.2 Scratch damage analysis of brittle PMMA 4.2.1 Scratch damage mechanisms of brittle PMMA 16 / 57

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Fig. 16a is the optical scanning image of a scratched PMMA sample. It can be found that, with the progressive increase of scratch depth, PMMA shows different scratch damage patterns, such as sliding indentation, periodic crack (Fig. 16b) and material removal, which is similar to the literatures (Cheng et al., 2016; Schirrer, 2005, 2011, Zhang et al., 2016).

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Fig. 16. Scratch damage phenomenon of PMMA: (a) optical scanning image of a scratched sample, (b) periodic crack.

Fig. 17 shows the experimental and numerical results of the normal force-scratch

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depth. It can be found that the scratch normal force response is sensitive to the scratch velocity. The numerical model can reasonably describe the velocity effect on the normal force-scratch depth curves of brittle PMMA using a constitutive framework

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similar to that of ductile PC.

Fig. 17. Comparison between experiment data and FEM results for PMMA under

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different scratch velocities.

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The evolutions of effective plastic stretch and craze strain during the scratch

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process of PMMA are shown in Fig. 18 and 19, respectively.

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Fig. 18. Evolution of effective plastic stretch during PMMA scratch at different scratch depth: (a) 0.01mm, (b)0.028mm, (c)0.0523mm, (d) 0.064mm.

Fig. 19. Evolution of craze strain during PMMA scratch at different scratch depth: (a) 0.028mm, (b) 0.052mm, (c) 0.0523mm, (d) 0.064mm.

Fig. 18a indicates that shear flow occurs firstly at the subsurface. With the progressive increase of scratch depth, λshear keeps increasing. The maximum λshear 17 / 57

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always occurs beneath the surface. This tendency is somewhat similar to that of ductile PC. From Fig. 18a and Fig. 19a, the occurrence shear flow (@scratch depth = 0.01mm) is earlier than that of craze flow (@scratch depth = 0.028mm). Fig. 19a shows that craze initiates at the top surface of substrate while the ductile fracture criterion is not satisfied yet. As required by craze initiation criterion, the maximum

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principle stress at the top surface reaches the critical value (Fig. 20a), as well as the positive mean stress appears at the same position (Fig. 20b). The similar scratch damage phenomenon for the brittle polystyrene, i.e., craze occurs at the surface, while shear-yielded zone occurs below, was also observed in previous experimental work

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(Xiang et al., 2001).

Meanwhile, after the initiation of craze flow at the surface, the magnitude of craze strain keeps increasing, and the region enduring craze flow expands with the increase of scratch depth. After reaching the brittle fracture criterion (Equation 10), a

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scratch-reduced crack now can be observed at the substrate behind the scratch tip (Fig. 19c). After the stress release caused by the brittle crack, the next crack will

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consequently occur at the position where the brittle fracture criterion is met again. Fig.

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18d shows the periodic pattern of scratch-induced cracks.

Fig. 20. Stress distribution at the moment of craze initiation: (a) maximum principle

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stress, (b) mean stress.

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The evolution process of the maximum effective plastic stretch and the maximum

craze strain during the scratch of PMMA is given in Fig. 21. It can be found that at the beginning of scratch, the effective plastic stretch develops faster than the craze strain. However, after the craze initiation, craze strain increases more quickly than the effective plastic stretch. When it reaches the critical value of brittle fracture, the scratch-induced craze-type crack occurs. After the occurrence of this crack, the effective plastic stretch still increases with the increase of scratch depth. 18 / 57

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Fig. 22 shows the stress release process caused by the craze-type crack. After the occurrence of crack, the maximum principle stresses at Location A and B decrease (Fig. 22b) from that of Fig. 22a. The stress release will stop the further propagation of crack in the depth direction. Meanwhile, it leads to a quick drop of the tangential force applied on scratch tip, which causes the stick-slip phenomenon. Drawing by the moving-forward scratch tip, the stress magnitude at the substrate behind the scratch

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tip (Location B) in Fig. 22b is larger than that of Location A. As the scratch tip continues to move tangentially, the magnitude of maximum principle stress behind the scratch tip will increase again. The craze-type crack occurs once more when the brittle fracture criterion is met. Ultimately, this will result into the periodic crack pattern (Fig

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22c), which is similar to the experimental observation (Fig. 16b). It should be noted that the stress release was also acknowledged as the main reason for the periodic crack pattern by both the analytical work (Bower et al., 1994) for an ideally brittle elastic material and the experimental results (Jiang et al., 2009) for the brittle

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polymers.

scratch.

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Fig. 21. Evolutions of max effective plastic stretch and max craze strain during PMMA

Fig. 22. Stress release phenomenon of the maximum principle stress: (a) before crack

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occurs; (b) after the crack occurs.

4.2.2 Effect of scratch velocity on the scratch damage for PMMA

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As shown in Fig. 23a, a decrease of scratch velocity can delay the occurrence of

scratch damage of PMMA. This effect of scratch velocity on scratch behavior has been well captured by the FEM simulation (Fig. 23b) using the constitutive model.

Fig. 23. Effect of scratch velocity on the scratch damage of PMMA: (a)experimental results; (b)FEM results

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Experimentally observed by Jiang et al. (2009) for soft TPO, a dramatic increase of scratch velocity can lead to the switch of scratch damage mechanism from a ductile mode to a brittle mode. For the purpose of contrary demonstration, the brittle PMMA is fictionally scratched at a very slow velocity (4e-6mm/s). With the same set of material parameters and simulation process as before, the ductile-type fracture due to shear flow occurs (Fig. 24a), while the corresponding craze strain is zero (Fig. 24b).

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Here, the constitutive model effectively describes the ductile-brittle transition of scratch damage of same material under different scratch velocity. As noted by Briscoe et al. (1996), the effective strain rate for scratch could be estimated as   v d , in which v is the scratch velocity and d is the scratch width. For the 1mm/s and

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100mm/s scratch velocity, the corresponding strain rates are roughly 1 s-1 and 102 s-1, which is higher than that of unaxial tests. Despite of this inconsistency, the general tendencies, such as a faster scratch speed bring out an earlier occurrence of craze-type

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failure, still follows.

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Fig. 24. Scratch deformation of PMMA at 4e-6mm/s: (a) effective plastic stretch, (b) craze strain

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For the scratch coefficient of friction, Van Breemen et al. (2012a, 2016) and Schirrer et al. (2005, 2008) have performed excellent works. In this work, the lateral

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force was not obtained due to the limitation of the scratch machine. Thus, no direct comparison of the lateral force between the FEM and experimental result was

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performed.

It should be noted that the effects of viscoelasticity (Schapery, 2000; Anand and

Ames, 2006, Brostow et al., 2006) on the scratch performance of polymers, as well as other scratch damage mechanisms (Bermúdez et al., 2005; Jiang et al., 2009, 2015), have not been investigated in this work. Further effort on the constitutive model and numerical simulation will be conducted for a better understanding of scratch behavior of polymers. The effect of scratch tip shape, as reported by many researchers (Briscoe 20 / 57

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et al., 1996; Van Breemen et al., 2012a), is another important concern which should be addressed in the further study of scratch damage mechanisms. 5. Concluding remarks In this work, to describe the competition between shear flow and craze flow, a crazing initiation criterion was introduced in an elastic-viscoplastic constitutive model. With the validated model, the FEM simulations was performed to investigate the

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scratch damage mechanisms of ductile PC and brittle PMMA.

(1) With the introduction of strain rate on the craze initiation criterion, a constitutive model capable of describing the competition between shear flow and craze flow has been proposed and validated;

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(2) For ductile PC, the shear flow-induced fracture occurs beneath the surface, propagates to the surface with the increasing of the scratch depth; (3) For brittle PMMA, while the shear flow occurs firstly at the subsurface, the craze initiates at the surface, propagates and breakdowns when the brittle fracture

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criterion is met.

(4) The stress release due to craze-type crack is one of the key factors for the periodic

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crack during scratch of PMMA. The effect of scratch velocity on the switch of scratch damage mechanisms for PMMA is also discussed.

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The above findings are helpful for a better understanding of the scratch damage mechanism of ductile PC and brittle PMMA, and could give a guidance for designing

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scratch-resistant amorphous polymers. Acknowledgements

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The authors thank the financial support from National Natural Science Foundation

of China (11472231) and Science and Technology Department of Sichuan Province (2013JQ0010). The authors also appreciate the constructive discussion with Dr. Yu Chao about the constitutive model. Appendix A. Determination of constitutive parameters In this appendix, the procedure to identify the material parameters in constitutive model is introduced. Firstly, the one-dimensional form of constitutive model is 21 / 57

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introduced. Then the procedure to estimate material parameters will be introduced. A.1. One-dimensional constitutive model In the one-dimensional constitutive model, Cauchy stress is expressed as σ . The stretch is U=U eU p , in which U e and U p are the elastic and plastic parts of stretch, respectively. The logarithmic strain is expressed as ε= ln U .

  



For craze initiation, σ1  σ1,cr  c1+

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The stress function in the one-dimensional form is σ=E ln U e =E ε-ε p .

3c2 +c3 ln(ε)  0 and σ  0 . σ





p p p U p d exp  ε  dt exp  ε  ε   εp Flow rule is U =D U . Then D = p  p p U exp  ε  exp  ε 

p

p

p

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p

If the craze initiation criterion is not satisfied, ε =εS . εSp  γ psign(σ-σback ) in p

p

 As   τ 5 6   which γ =γ0 exp - 1-     with τ= σ-σback . The pressure effect is not  θ   s    p

plastic stretch.

1

1

 λp  p 2 p-1 1 U p2 +2U p-1 is the effective   U -U  , in which λp = 3  λL 

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 λ  σback =μR  L   3λ   p

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considered in the one-dimensional equations. The evolution of back stress is

 x  is the inverse of Langevin function. The evolution function of

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s p   s=h( 1- s )γ the internal variables are  , s=scv 1+b  ηcv -η   . The criterion for η=g0 ( s -1 )γ p scv 

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shear flow induced fracture is λp  λpf . If the craze initiation criterion is satisfied, ε =εC with εCp  ξ p sign(σ1 ) . The p

p

f craze strain is εcraze = ξ p dt . When the craze strain reaches a critical value, εcraze  εcraze ,

craze breakdown occurs. A.2. Identification of material parameters The material parameters to be identified are: 22 / 57

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(1) Reference Young’s modulus E0 at strain rate ε0 , Poison’s ratio υ and strain rate sensitivity parameter k E . The minimum strain rate is taken as ε0 in uniaxial compression test and E0 is the corresponding Young’s modulus. Poison’s ratio υ is estimated from υ=- εtrans εaxial ,

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where the strain along the loading direction εaxial and the strain vertical to the loading direction εtrans can be obtained from a uniaxial compression test at ε0 using the DIC

the Young’s modulus at strain rate ε1 ;

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E  1 test data. k E can be determined according to k E =  1 -1 , in which E1 is  E0  log  ε1   ε0 

(2) Parameters in the shear flow part: Rubbery modulus μR , network locking stretch

λL ,  0 , A,  in the rule of shear flow, as well as h, b, g0 , s0 scv , ηcv  in the

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function of internal variables.

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The pressure sensitivity parameter can be identified as α = 3  C  T   C  T  , here, C and T are the tensile and compressive yield strength at the evaluated strain rate,

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respectively.

Assuming that the deformation resistance s does not evolve (Anand and Ames,

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 As   τ 5 6   2006), that is s=s0 . From A.1, ε  γ0 exp   0 1      sign(σ-σback ) .  θ   s0      p

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Considering the uniaxial tension condition, sign(σ-σback )  1 , τ= σ-σback =σ-σback . Then we can obtain

 As   σ-σ 5 6   εp  exp   0 1   back    γ0  θ   s0     

(11)

56  εp  As0   σ-σback   1   ln         γ θ s  0   0  

(12)

Rearrange the above formula as

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Then we can have 6

  εp θ σ  1  ln   As0  γ0

 5   s0 +σback 

(13)

Under a large deformation, the effect of elastic deformation is not as significant p e as plastic deformation. Assuming ε =ε-ε =ε- σ

E

 ε , Equation (13) can be

6

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rewritten as   ε  5 θ σ  1  ln    s0 +σback  As0  γ0  

Adopting an approximation of the inverse of Langevin function

σ back

1

 x  x

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the back stress in A.1 can be rewritten as

(14)

2  λL   λp 3-  λp λL   p 2 p-1   U -U   μR  2   3λ   λ  p   L 1-  λp λL  

(15)

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2   μR  3-  λp λL   p 2 p-1 = U -U  3  1-  λ λ 2  p L  

3-x 2 , 1-x 2

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Substituting Equation (15) into Equation (14), one can get 2     ε  5 μR  3-  λp λL   p 2 p-1 θ σ  1  ln    s0  U -U  2    As γ 3 0  0    1-  λp λL   6

PT

(16)

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The strain rate sensitivity is only concerned with the first term on the right-hand side of Equation (16), while the strain hardening term is only related to the second

ε1 ,ε1 ,σ1 , ε1 ,ε2 ,σ 2  and

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term on the right-hand side. From Fig. 25, three points

ε1 ,ε3 ,σ3 from experimental results at one fixed strain rate are used to determine

rubbery modulus μR , network locking stretch λL according to Equation (17) and (18).

Fig. 25. Illustration to determine μR , λL , s0 and A.

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2 2      μR  3-  λp 2 λL   p 2 p-1  3-  λp1 λL   p 2 p-1  σ 2 -σ1= U -U1  U 2 -U 2  -  2  1   3  1-  λ λ 2  1 λ λ   p2 L p1 L     

(17)

2 2      μR  3-  λp 3 λL   p 2 p-1  3-  λp1 λL   p 2 p-1  σ3 -σ1= U -U1  U3 -U3  -  2  1   3  1-  λ λ 2  1 λ λ p3 L    p1 L    

(18)

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Similarly, s0 and A can be estimated at the same strain level at three different strain rates (see in Fig. 25) according to Equation (19) and (20). 6

6

   ε  5  ε  5 θ θ σ 4 -σ1= 1+ ln  2   s0 - 1+ ln  1   s0  As0  γ0    As0  γ0   6

(19)

6

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   ε  5  ε  5 θ θ σ5 -σ1= 1+ ln  3   s0 - 1+ ln  1   s0  As0  γ0    As0  γ0  

(20)

The material constants h, b, g0 , scv are curve-fitted to control the shape for the yield peak. h controls the pre-peak slope, while b, g0 , scv affect the yield peak and

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Gearing and Anand, 2004).

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post-yield response [46]. ηcv is obtained from the literature (Anand and Gurtin, 2003;

(3) Parameters in the craze flow part: c1，c2 , c3 , scraze , m and fcraze

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c1 ,c2 ,c3 are determined from the uniaxial test with different strain rates under different strain rate. The rate-sensitivity parameter m is estimated from





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m= ln σ y 2 σ y1 ln  ε2 ε1  , in which σ y1 and

σ y2

are the yield stresses corresponding

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to the strain rates ε1 and ε 2 , respectively. scraze is estimated from the same method with Gearing and Anand (2004). ξ 0 is obtained from the following Equation (21) to guarantee the continuity of plastic stretching when the material flow switches from shear to craze. In Equation (21), a quantity with a superscript (  ) denotes its value at the f time when the change of material flow is activated. εcraze is determined by fitting the

uniaxial tensile strain-stress curve till its fracture point. 25 / 57

ACCEPTED MANUSCRIPT  A  s* -ασ*m    τ * 5 6    s m craze  1-  *  ξ0 =γ0 exp    s -ασ*m     σ1*  θ     1

(21)

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Richeton, J., Ahzi, S., Vecchio, K., Jiang, F., Makradi, A., 2007. Modeling and validation of the large deformation inelastic response of amorphous polymers over

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Richeton, J., Schlatter, G., Vecchio, K.S., Rémond, Y., Ahzi, S., 2005. A unified model for stiffness modulus of amorphous polymers across transition temperatures and

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strain rates. Polymer 46, 8194-8201. Schapery, R.A., 2000. Nonlinear viscoelastic solids. International Journal of Solids

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Zhang, J., Jiang, H., Jiang, C., Cheng, Q., Kang, G., 2016. In-situ observation of temperature rise during scratch testing of poly (methylmethacrylate) and

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Figure

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Fig. 1. The geometry of the specimen for tensile tests and scratch tests.

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Fig. 2. (a) The geometry of the specimen of PMMA thin plate with a circular hole,

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(b) Finite element mesh near the hole.

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Fig. 3. Schematic plot of finite element model for scratch simulation.

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Fig. 4. A one-dimensional rheological representation of the proposed constitutive

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model considering shear yielding.

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(b) 200

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Fig. 5. Comparison between experiment data and predicted curves of PC under

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different strain rates: (a)tension, (b)compression.

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90 Exp

(b) 250

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60 16

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Stress (MPa)

Stress (MPa)

1e-2/s 1e-3/s 1e-4/s

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Fig. 6. Comparison between experiment data and predicted curves of PMMA under

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different strain rates: (a)tension, (b)compression.

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100

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Fig. 7. Strain rate effect on the craze initiation of PMMA.

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Fig. 8. Effect of displacement rate on the force-displacement curve of PMMA plate

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with a circular hole under tension.

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Fig. 9. Comparison between DIC and FEM results of strain distribution along the

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vertical.

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Fig. 10. Comparison between DIC and FEM results of strain distribution along the

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tensile direction under 0.005mm/s: (a) DIC result; (b) FEM result. Loading direction is

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vertical.

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Fig. 11. Contour plots of εcraze : (a) craze initiates, (b) before crack occurs, (c) crack

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occurs, (d) specimen separates into two parts. Loading direction is vertical.

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Fig. 12. Scratch damage phenomenon of PC: (a) optical scanning image of a scratched

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sample, (b) microscopic images of scratch damage.

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Fig. 13. Comparison between experimental normal force-scratch depth curves and

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FEM results for PC scratch.

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Fig. 14. Evolution of effective plastic stretch during PC scratch at different scratch

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depth: (a) 0.011mm, (b)0.094mm, (c)0.0943mm, (d)0.105mm.

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(a)

(b)

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Fig. 15. Scratch induced fracture of PC: (a) FEM result, (b) corresponding test data.

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Fig. 16. Scratch damage phenomenon of PMMA: (a) optical scanning image of a

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scratched sample, (b) periodic crack.

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Fig. 17. Comparison between experiment data and FEM results for PMMA under

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different scratch velocities.

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Fig. 18. Evolution of effective plastic stretch during PMMA scratch at different scratch 49 / 57

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depth: (a) 0.01mm, (b)0.028mm, (c)0.0523mm, (d) 0.064mm.

Fig. 19. Evolution of craze strain during PMMA scratch at different scratch depth: (a) 0.028mm, (b) 0.052mm, (c) 0.0523mm, (d) 0.064mm.

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Fig. 20. Stress distribution at the moment of craze initiation: (a) maximum principle

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stress, (b) mean stress.

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Fig. 21. Evolutions of max effective plastic stretch and max craze strain during PMMA

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scratch.

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occurs; (b) after the crack occurs.

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Fig. 22. Stress release phenomenon of the maximum principle stress: (a) before crack

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(b)

Fig. 23. Effect of scratch velocity on the scratch damage of PMMA: (a)experimental

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craze strain

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Fig. 24. Scratch deformation of PMMA at 4e-6mm/s: (a) effective plastic stretch, (b)

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Fig. 25. Illustration to determine μR , λL , s0 and A.

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Table

Table 1 Material parameters for PC Elastic parameters: E  2.2GPa , v=0.33 , kE =0.021 Parameters for shear flow part: μR  9MPa , λL  1.67 , γ0  5 104 / s ,

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A=460K/MPa , θ  293K , α =0.2 , h=1.75GPa , s0  18MPa , scv  24MPa , b=790 ,

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g0  8 103 , ηcv  0.001 , λpf  1.33

Table 2 Material parameters for PMMA Elastic parameters: E  2.8GPa , v=0.33 , kE =0.053

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Parameters for shear flow part: μR  7.7MPa , λL  1.71 , γ0  5 104 / s , A=167K/MPa , θ  293K , α =0.32 , h=1.1GPa , s0  20MPa , scv  31.7MPa ,

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3 b=790 , g0  6 10 , ηcv  0.00025 , λpf  1.33

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Parameters for craze flow part: c1  0.2MPa , c2  785.56MPa 2 , c3  46.7MPa  s ,

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f scraze  200MPa , m=0.05 , εcraze  0.005

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