J. Cent. South Univ. Technol. (2008) 15(s1): 555-559 

DOI: 10.1007/s11771-008-420-1 

Influence of strain rate on fracture behavior of poly(methyl methacrylate)

ZHAO Rong-guo(赵荣国)^{1, 2}, CHEN Chao-zhong(陈朝中)¹

(1. Institute of Rheological Mechanics, Xiangtan University, Xiangtan 411105, China;

2. Key Laboratory of Low Dimensional Materials and Application Technology of Ministry of Education,

Xiangtan University, Xiangtan 411105, China)

Abstract: The effect of strain rate on fracture behavior of poly (methyl methacrylate) was investigated. The uniaxial tensile rupture tests for the poly (methyl methacrylate) samples were carried out at different strain rates at ambient temperature. It is found that the elastic modulus of the material increases with increasing strain rate, while the elongation is reversal with strain rate. Simultaneously, there exists a critical strain rate within which the stress-strain curves overlap one another, and beyond which the curves depart from each other. The amount of energy added to the system due to work done by the imposed load was calculated, and the strain energy stored in the material at each strain rate was calculated by the current stress integral with respect to strain. The complementary strain energy, which is the difference between the work and the strain energy, was obtained and was considered to supply the surface energy to create a new crack surface in the polymeric material. It is found that the work done by the imposed load, which is needed for the fracture of poly (methyl methacrylate) sample, decreases with increasing strain rate, and the strain energy decreases with strain rate as well, which demonstrates that the polymeric material at high strain rate is easier to fracture than that at low strain rate. As the strain rate increases, the fracture mode changes from ductile, semi-ductile to brittle mode. The complementary strain energy almost sustains a constant at any strain rate. The density of surface energy, which characterizes the energy per unit area needed for creating crack surface, is a strain rate-independent material constant.

Key words: viscoelasticity; fracture; strain energy; complementary strain energy; polymer

1 Introduction

The polymers and polymer-matrix composites have been widely used as load-carrying components in many engineering structures. It is known that one of the weaknesses of plastics is their poor impact resistance. Even the most ductile polymers can break in a brittle manner under impact load^[1]. Therefore, a thorough knowledge about their mechanical properties and failure behaviors is necessary. The various viscoelastic constitutive models that were presented to describe the rheologic behaviors of polymers, are reviewed in this work, and the effect of strain rate on fracture behavior of poly (methyl methacry-late) is analyzed.

2 Viscoelastic constitutive models

2.1 Linear viscoelastic constitutive models

In linear viscoelastic theory, the rheologic behaviors of polymers are characterized using the differential form with mechanical analogy of springs and dashpots or the single integral representation. In terms of the Boltzmann superposition principle, the creep type and relaxation one of constitutive equation are individually written as

                    (1)

                    (2)

where  ε(t), σ(t), D(t) and E(t) are strain, stress, creep compliance and relaxation modulus, respectively.

2.2 Multiple integral constitutive models

Linear viscoelastic theory has been widely used for polymers at low stress levels. But for polymers at high or intermediate stress levels, the nonlinear viscoelasticity appears. To describe nonlinear viscoelastic behavior, the multiple integral constitutive equations are presented as

  (3)

 (4)

where  the first term denotes the linear response, and the second term is the strain (stress) amount derived by the mutually action of two stress (strain) increments. D_i(t), E_i(t) (i=1, 2, …) are individually creep compliance and relaxation modulus.

2.3 Single integral constitutive models

The multiple integral constitutive equations are of integral multinomial containing infinite integral terms, in fact, they can precisely describe the material’s nonlinear viscoelasticity. Whereas, there are so many material functions in the representations that complex arises from experiments and calculations even only a few terms are selected. Thus, many nonlinear viscoelastic models in the form of single integral were presented and developed.

A nonlinear stress (strain) function, instead of stress (strain) history in the Boltzmann superposition principle, was introduced into the form of convolution integrals of linear viscoelasticity, leading to^[2]

         (5)

        (6)

where  D₀ is the initial creep compliance; E₀ is the initial relaxation modulus; g[σ(t)] and f[ε(t)] are the empirical functions of stress and strain, respectively.

Based on the irreversible thermodynamics, a non- linear viscoelastic representation in the form of single integral was proposed, and the equation in creep type is^[3]

    (7)

where

are stress-reduced times; a_σ is a time scale factor; g₀, g₁, and g₂ are all the functions of stress, g₀ represents the linearity of the instantaneous response, g₁ reflects the multiplier of heredity integral, and g₂ takes the load rate effect on the strain response into consideration. Similarly, the equation in relaxation type is

     (8)

where

are strain-reduced times; a_ε is a time scale factor; h_∞, h₁, and h₂ are all the functions of strain; h_∞E_∞ is the equilibrium state; h₁ denotes the increment of heredity integral; and h₂ states the strain rate effect on the stress response.

2.4 Constitutive models based on elasticity recovery

Let x, σ(t) and ε(t) be the position, the stress and the strain in one-dimensional case, respectively, suppose that the viscoelastic material possesses instantaneous elastic response. Let σ^e(t) and ε^e(t) be the instantaneous elastic stress and strain, respectively. If a material obeys the first and the second laws of thermodynamics, and the load process is reversible, then the material possesses elastic property. It follows that there exists a strain energy function, U_E=U_E(ε^e, x, t) with the property that

                              (9)

and a complementary strain energy function U_c=-U_E+σ^e ε^e=U_c(σ^e, x, t) with the property that

                             (10)

For an isotropic viscoelastic material with memory property, the equations of the current strain (strain) and the instantaneous elastic strain (stress) are written as^[4-6]

           (11)

          (12)

where nondimensional quantity D(t)=D_ε(t)/D_ε0 denotes the relative creep compliance; E(t)=E_σ(t)/E_σ0 is referred to the relative relaxation modulus, and the symbol “*” denotes Stieltjes convolution. The creep compliance D(t) and relaxation modulus E(t) satisfy D*dE=E*dD=H(t), where H(t) is the Heaviside function. The converse expressions of Eqns.(11) and (12) are given as

            (13)

          (14)

where the recovered strain ε^e(t) and stress σ^e(t) solved from current strain ε(t) and stress σ(t) via the hereditary integral are called the recovered instantaneous elastic strain and stress.

If ε(t) is understood as a prescribed input quantity, then the known strain itself is the instantaneous strain, i.e. ε(t)=ε^e(t). In this case, the recovered elastic stress σ^e(t) is calculated by Eqn.(14), the instantaneous elastic constitutive equation, σ^e(t)=φ[ε(t)], is derived and then substituted into Eqn.(12) to predict the current stress response. In the case of σ(t) known as an input quantity, i.e. σ(t)=σ^e(t), the recovered elastic strain ε^e(t) is simulated by Eqn.(13), the instantaneous elastic constitutive relation, ε^e(t)=Ψ[σ(t)], is deduced and then introduced into Eqn.(11) to forecast the strain response.

3 Fracture energy

For a certain solid material with crack, if the change of energy produced by the small change in crack system is taken into account, then the basic equation describing the crack propagation condition can be deduced in terms of the law of energy conservation in thermodynamics or in the classic mechanics. To put the energy analysis, it is significant to look at the schematic diagram of the crack propagation condition in Fig.1. In Fig.1(a), a load P acts on the boundary of a viscoelastic body, and the internal energy of the system is U_A. The case shown in Fig.1(b) is similar to that in Fig.1(a), but a crack with length 2c is introduced into the body. The mechanical flexibility of viscoelastic body will decrease as the result of existence of a crack, and a small deformation of the body occurs under the imposed load P.

Fig.1 Schematic diagram of crack propagation

In the next, the change of the total internal energy of the viscoelastic body corresponding to the state without crack and that with crack are investigated. Firstly, the crack produces a new surface, which results in an increment U_S, that is, the surface energy of the system. Secondly, due to the small difference of the viscoelastic body’s shape, the position of the imposed load P changes, the amount of the work due to the load to the body is W. Finally, when the body contains a crack, the increment of the strain energy stored in the body will be U_E. Thus, the total internal energy of the system in Fig.1(b) can be written as^[7]

                    (15)

For the real reversible thermodynamic system, the work of internal force is equivalent to the loss of the system’s potential energy, so the amount of the work is negative.

Suppose that the crack propagates a small distant δc under the action of load P, as shown in Fig.1(c), and the total internal energy is

                          (16)

The thermodynamic theory suggests that the crack propagation does not result in increasing the internal energy, so the sufficient condition of crack propagation is

≤0                                  (17)

Simultaneously, the thermodynamic theory indicates that the process that reduces the total energy of the system can spontaneously occur, therefore, Eqn.(17) is also the necessary condition of crack propagation.

Eqn.(17) is the energy balance criterion to predict the material’s fracture behavior. In the theoretic point of view, if only the special expressions of all terms in Eqn.(11) are given, the basic equation of crack propagation can be derived. The equality in Eqn.(17) corresponds to the equilibrium state of the body with crack, following

                               (18)

In this case, the complementary strain energy U_c, i.e. W-U_E, is used to supply the surface energy to create a new crack surface.

4 Effect of strain rate on fracture behavior

The uniaxial tensile rupture tests for poly (methyl methacrylate) were carried out on a CSS44020 Universal Electron Test Machine at different strain rates varying from 4×10^-6 s^-1 to 4×10^-4 s^-1 at ambient temperature. The samples were machined from a poly (methyl metha-crylate) sheet and made into the shape of dumbbell with dimensions 10 mm×2 mm×60 mm in the gauge zone. The stress vs strain curves are shown in Fig.2. It is found from Fig.2 that the strain responses are dependent on the strain rates, and the elastic modulus increases with increasing strain rate, while the elongation is reversed with strain rate. Simultaneously, it can be seen from Fig.2 that the curve at strain rate 8×10^-6 s^-1 is almost overlapped by the one at strain rate 4×10^-6 s^-1, so there exists a critical strain rate within which the stress-strain curves overlap one another so the response is of strain rate-independent, and beyond which the curves depart from each other so the response is of strain rate-dependent.

Fig.2 Stress vs strain curves of poly (methyl methacrylate)

The amount of energy added to the system due to work done by the imposed load P is calculated for each strain rate. Following that, the stress vs strain curves at different strain rates are simulated via the nonlinear viscoelastic constitutive equations in the form of single integral aforementioned above to forecast the current stress response. The strain energy U_E stored in the body at each strain rate is calculated by the current stress integral with respect to strain, i.e.

                        (19)

The complementary strain energy U_c, which is the difference between the work and the strain energy, i.e. W-U_E, is then calculated and is considered to supply the surface energy to produce a new crack surface in the viscoelastic body. By the way, the complementary strain energy U_c can also be calculated by the strain integral with respect to stress, i.e.

                        (20)

The work done by the imposed load P vs lg(dε/dt) curve is plotted in Fig.3. It is found from Fig.3 that the amount of energy added to the material due to the work done by the imposed load, which is needed for the fracture of the poly (methyl methacrylate) sample, decreases with increasing strain rate, which demonstrates that the polymeric material at high strain rate is easier to fracture than that at low strain rate. As the strain rate increases, the fracture manner of polymer changes from ductile to semi-brittle and then to brittle manner.

Fig.4 shows the strain energy vs logarithmic strain rate curve. It can be found from Fig.3 that the strain energy is reversal with strain rate. The tendency of the strain energy curve is similar to that of the work curve.

Fig.3 Work done by load P vs strain rate curve

Fig.4 Strain energy vs strain rate curve

The complementary strain energy vs logarithmic strain rate curve is shown in Fig.5. It is found that the complementary strain energy almost sustains a constant at any magnitude of strain rate. As aforementioned above,

Fig.5 Complementary strain energy vs strain rate curve

the complementary energy is considered to supply the surface energy to form a new crack surface. Note that

                   (21)

When the sample completely fractures, the fracture surface is two times of the area of cross section. Here, A=20 mm². According to Eqn.(21), the surface energy density γ_S can be determined. Obviously, γ_S is a constant for a specified polymeric material. It characterizes the energy per unit area needed for creating crack surface.

The work done by the imposed load and the strain energy are reversal with strain rate, which demonstrates that the polymeric material at high strain rate is easier to fracture than at low strain rate. The surface energy density is a strain rate-independent material constant.

[1] VU-KHANH T. The impact fracture of polymers: unanswered questions [J]. Trends in Polymer Science, 1997, 5(11): 356-360.

[2] FINDLEY W N, LAI J S. A modified superposition principle applied to creep of nonlinear viscoelastic material under abrupt changes in state of combined stress [J]. J Rheol, 1967, 11(3): 361-380.

[3] SCHAPERY R A. On the characterization of nonlinear viscoelastic materials [J]. Polym Eng Sci, 1969, 9(4): 295-310.

[4] ZHANG C, ZHANG W. Elasticity recovery correspondence principles for physically nonlinear viscoelastic problems for a class of materials [J]. Int J Solids Struct, 2001, 38(46/47): 8359-8373.

[5] ZHAO Rong-guo. Novel measuring approach for damage of viscoelastic material (Part I): Constitutive model [J]. J Cent South Univ Technol, 2007, 14(s1): 289-292.

[6] ZHAO R G, LUO W B. Time-dependent mechanical behaviors of polyamide 6/nano-SiO₂ composite [J]. Key Eng Mater, 2008, 368-372: 1080-1083.

[7] GRIFFITH A A. The phenomena of rupture and flow in solids [J]. Philos T Roy Soc A, 1921, 221: 163-198.

(Edited by YUAN Sai-qian)

Foundation item: Projects(10772156, 10672136) supported by the National Natural Science Foundation of China

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: ZHAO Rong-guo, Associated Professor; Tel: +86-732-8293214; E-mail: zhaorongguo@xtu.edu.cn