The strain gradient viscoelasticity full field solution of mode-III crack problem

The size and viscosity effects are noticeable at the micro-/nano scale. In the present work, the strain gradient viscoelastic solution of the mode-III crack in an infinite quasi-brittle advanced material is proposed based on the strain gradient viscoelasticity theory using the Wiener–Hopf method. The solutions to the gradient-dependent viscoelastic crack problem are obtained directly by using the correspondence principle between the strain gradient viscoelasticity and strain gradient elasticity in Maxwell’s standard linear solid model. In this model, the stress near the crack tip is time-dependent and size-dependent. Besides, the stress near the crack tip is more significant than that based on gradient elasticity theory. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. The time that the normalized stress takes to stabilize also changes with the distance from the crack tip. When the viscosity effect is neglected or time tends to infinity, the strain gradient viscoelasticity theory can be reduced to the classical strain gradient elasticity theory.


Introduction
Compared to standard continuum solid mechanics theories, gradient theories can capture the sizedependent mechanical response of materials. Therefore, different forms of gradient theory have been developed to interpret the size-dependent material properties differently. In the 1960s, Mindlin (1964) developed the general strain gradient elasticity theories including several extra material parameters that are difficult to determine by experiments. As a landmark work, many subsequent research works were completed on this basis. Lam et al. (2003) indicated that the potential energy only depends on strain tensors and the symmetric part of the rotation gradient tensor. Thus, a modified strain gradient elasticity theory was proposed, and the number of length scale parameters was reduced to three. Based on the strain gradient theory proposed by Mindlin et al. (Mindlin 1964;Mindlin and Eshel 1968), Aifantis et al. (Aifantis 1992;Altan and Aifantis 1997) considered strain gradients in the elastic range and developed a simplified strain gradient elasticity theory with only one scale parameter. Then, Gao and Park (Gao and Park 2007) presented the variational formulation of simplified strain gradient elasticity theory using the minimum total potential energy principle and presented a more complete boundary condition. In addition, the couple stress elasticity has been developed. The traditional couple stress theory, using macro-rotation as the accurate kinematical rotation, was developed by Toupin and Mindlin et al. (Mindlin and Tiersten 1962;Toupin 1962;Mindlin 1963). Moreover, two additional material parameters are introduced besides the Lame constants. It is difficult to obtain these material parameters because the spherical part of the couple-stress tensor is hard to determine, and the body couple exits in the constitutive relation for the force-stress tensor. Yang et al. (2002) proposed a modified couple stress elasticity theory, which introduced an equilibrium condition of moments of couples. Thus, the couple stress tensor is symmetric, and only one internal material length scale parameter is involved. Meanwhile, many related research works have been done, and different modified couple stress elasticity and strain gradient elasticity models have been derived (Akgöz and Civalek 2011;Hadjesfandiari and Dargush 2011;Ş imşek and Reddy 2013;Lim et al. 2015). Gradient elasticity theories have been used to solve many problems, like beam bending (Papargyri-Beskou et al. 2003;Lazopoulos and Lazopoulos 2010;Lurie and Solyaev 2018), vibrations of beams (Reddy 2007;Al-Basyouni et al. 2015), micro-/nano-indentation (Huang et al. 2000;Wei and Hutchinson 2003;Zhao et al. 2003), and cracks (Sciarra and Vidoli 2013;Mousavi et al. 2014a;Joseph et al. 2018).
The crack problem is a classical scientific problem. Many studies have carried out detailed research on this problem (Vardoulakis et al. 1996;Moës and Belytschko 2002;Wang et al. 2008;Kaminsky et al. 2014;Mousavi et al. 2014b). Gradient theories have been widely used to analyze the mode-III crack problem. Chen et al. (1997) derived a full field solution for the mode-III crack in an infinite medium using the Wiener-Hopf method. Georgiadis (Georgiadis 2003) analyzed mode-III crack in an infinite medium using strain gradient elasticity theory in statics and dynamics. Also, a stress singularity that is a departure from classical fracture mechanics was noted. Li and Wang (Li and Wang 2018) studied a mode III crack in an elastic layer on a substrate and solved it with both volumetric and surface strain gradient taken into account. Many other studies also solved different crack problems using strain gradient theories (Askes and Sluys 2003;Radi and Gei 2004;Fu et al. 2008;Donà et al. 2014;Lurie and Vasiliev 2019). Meanwhile, the viscoelastic effect in the crack problem has been studied. For example, Persson and Brener (2005) studied crack propagation in linear viscoelastic solids and proposed formulations for the velocity-dependent crack-tip radius based on energy conservation. To avoid the computational effect of the inverse Laplace-Carson transform, Nguyen (2014) introduced a generalized Kelvin viscoelastic model to study the viscoelastic properties of materials with random orientation distributed micro-cracks. Yao and Li (Yao et al. 2018) numerically solved viscoelastic crack problem by coupling the Symplectic Analytical Singular Element with a precise time-domain expanding algorithm.
The viscosity and size effects are evident at the micro-/nano-scale (Li et al. 2010;Yamada et al. 2013;. However, to the best of authors' knowledge, previous research work generally focused only on one of the effects and ignored the other when solving the mode-III crack problem. Therefore, the purpose of this study is to study the viscous effect and strain gradient effect of the mode-III crack problem using the strain gradient viscoelasticity theory.
In Maxwell standard linear strain gradient viscoelasticity solid model, the stress near the crack tip first rise and then fall and is more significant than that which is in gradient elasticity theory. The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. We will explain this phenomenon in Part 4.

Strain gradient viscoelasticity theory
For gradient-dependent, linear viscoelastic and isotropic materials, the total potential energy of the strain gradient viscoelasticity theory is written as Christensen (Christensen 2012; Lin and Wei 2020) where * is the Stieltjes convolution symbol and the same as follows. f i , p i and q i are body forces, surface tractions, and double tractions, respectively, and they are all functions of time. The normal derivative operator D is defined as D n i o i . The stress r ij t ð Þ and strain e ij t ð Þ are written as Christensen (Christensen 2012) where G t ð Þ and k t ð Þ, similar to the Lame constants in elastic theory, are the relaxation functions in viscoelasticity. The double stress s ijk is work-conjugated to the strain gradient e ijk . e ijk Is defined as And the double stress is written as where A s t ð Þ and A p t ð Þ are high-order relaxation functions. By analogy with classical gradient elasticity, these high-order material parameters should be related to the conventional relaxation functions in viscoelasticity. Thus, these two parameters are taken as where c t ð Þ is the gradient parameter, which is the representation of material's microstructure. Thus, the formulation of double stress in Eq. (5) can be updated Taking the first variation of Eq. (1), Applying 'Gauss' divergence theorem and using Eqs. (3, 4), the first term on the right-hand side of Eq. (8) can be written as Define the total stress l ij as Hence, Eq. (8) can be written as where the surface gradient operator D j is Let dP ¼ 0 the differential equilibrium equation can be obtained together with the boundary conditions l ij n j À D j ðn k s ijk Þ þ D l n l ð Þn k n j s ijk ¼p i or u i ¼û i ð14Þ s ijk n j n k ¼q i or u i;l n l ¼ oû i on ð15Þ The format of governing Eqs. (13) and boundary conditions derived here are similar to those given by Altan et al. (Altan and Aifantis 1997) and Gao et al. (Gao and Park 2007). However, it should be mentioned that variables in Eqs. (13-15) are the function of time. When the viscosity effect is ignored, this strain gradient viscoelasticity theory will reduce to the classical strain gradient elasticity theory.

Solutions of mode-III crack
In the present section, we focus our attention on the stationary mode-III crack in linear viscoelastic solids. Firstly, the solution of stress and strain are derived by using the equilibrium equation and Laplace transformation. Subsequently, solutions of the gradient-dependent viscoelastic crack problem can be directly obtained from the solution of the corresponding gradient-dependent elastic problem.

Full field solutions in strain gradient theory
The nonzero stresses in mode III crack are where w is the displacement. The higher stresses in mode III crack are as Wei and Gao (Wei 2006;Gao and Park 2007) We get the governing equation: The surface traction t k and higher surface traction b k are where s ij is Cauchy stress c 0 ¼ l 2 ; l is the characteristic length scale. For mode III crack, the boundary condition is: (a) Traction-free condition Anti-symmetry of anti-plane shear deformation: Because the asymptotic solution in strain gradient theory is non-physical (Chen et al. 1997), we solve Eq. (3) by using the Wiener-Hopf method. These are called Full-field solutions. The Fourier transform of a general function is Taking the Fourier transform of Eq. (18). The governing equation: Its bounded solution in the upper half plane is where Boundary (5) and (8) are transformed to (13) leads to We have The Fourier transform of boundary condition (20), which only applied to the negative half x axis gives where Similarly, the boundary condition (7) gives where Elimination of B 1 ðsÞ them lead to where The function kðsÞ can be factorized by The function RðtÞ is defined by The Eq. (36) can then be arranged to Using Liouville's theorem, we get Equation (42) then becomes And Surface traction can be obtained by inverse Fourier transformation Similarly, the shear stress The displacement is The higher stresses We get the full-field solutions in strain gradient elasticity theory.
3.2 Solutions in strain gradient viscoelasticity theory K III ¼ K III t ð Þ is the stress intensity factor, which is a function of time in strain gradient viscoelasticity theory, defined as where r 1 t ð Þ is the far-field anti-plane shear stress, and a is the half crack length. For viscoelastic problem, constant force loads the viscoelastic body.r 0 , that is r 1 ¼ r 0 H t ð Þ and HðtÞ is Heaviside function. Thus, the Laplace transformation of the stress intensity factor K III is The strain gradient viscoelastic solution of mode-III crack is obtained. In the following, as an example, the standard 3-parameter viscoelastic model is used to obtain formulations of the displacement and strain.
In Fig. 1, f is the higher-order dashpot. A 0 and A 1 are higher-order elastic components, which represent the higher-order modulus. The relations between higher-order modulus and traditional modulus are the same as the ones in the strain gradient elasticity theory.
where c e is the gradient parameter in strain gradient elasticity theories without considering a viscous effect, G 0 and G 1 denote the traditional elastic modulus. Thus, relations between the double stress and strain gradient of the higher-order viscoelastic model can be given as Applying the Laplace transform, the higher-order stress-strain relation can be obtained where A t ð Þ is defined as the high-order relaxation modulus of isotropic solids, A s ð Þ is in Laplace space, and the definition is similar to Eq. (59) Applying the inverse Laplace transform, the higherorder relaxation modulus is given where Fig. 1 The higher-order3-parameter viscoelastic model where j g is defined as the relaxation time of higherorder viscoelastic model, corresponding to a specific relaxation time of the Maxwell unit at the macronanoscale. The value of j g should be much smaller than s g of the Maxwell unit at macroscopic scale. The standard 3-parameter model is presented as an example to obtain the formulation of viscoelastic solutions, as shown in Fig. 1. The parallel combination of a spring and a dashpot is an important building block, called the Kelvin-Voigt unit. This 3-parameter model is obtained by adding a spring in series to a Kelvin-Voigt unit.
The creep function of this 3-parameter viscoelastic model in the Laplace space is where s g ¼ g=G 1 denotes the relaxation time. And the corresponding Laplace transformed relaxation function is (Christensen 1982) Furthermore, in order to obtain the concrete formulation of the viscoelastic gradient parameter, the relation between double stress and strain gradient must be presented. By analogy with the classical viscoelastic model, a high-order three-parameter viscoelastic model is introduced, as shown in Fig. 1.
Similarly, the higher-order creep function of this high-order 3-parameter viscoelastic model in the Laplace space can be obtained where j g ¼ f=A 1 denotes the high-order relaxation time. And the corresponding Laplace-transformed high-order relaxation function is Using Eqs. (6), (59), (60), (61) and (62), the Laplace transformed gradient parameter can be obtained The inverse Laplace transformation of Eq. (63) can be obtained by using the residue theorem In our previous study , we summarized a correspondence principle between the strain gradient viscoelasticity and the classical strain gradient elasticity. And the applicable conditions of this correspondence principle are the same as those of the traditional correspondence principle. In the 1960s, Graham (Graham 1968) proposed an extension of the traditional correspondence principle, which is suitable for mixed boundary value problems such as cracks and indentation. And the traditional correspondence principle has been used in many research works to solve the viscoelastic crack problems (Schapery 1984;Wang et al. 2014;Yao et al. 2018). Thus, the strain gradient viscoelastic solutions of the mode III crack problem can be obtained directly using our correspondence principle . Using our correspondence principle, the strain gradient viscoelastic solution in Laplace phase space can be obtained by replacing G with sG s ð Þ and c 0 with sc s ð Þ we get Then, using the standard 3-parameter viscoelastic model and the inverse Laplace transform, the components of stresses can be obtained. However, it should be noted that this correspondence principle might not be applicable when the crack propagates because the boundary conditions change.
Equation (70) can be obtained by applying the inverse Laplace transformation. It should be mentioned that the inverse Laplace transformed results of stress and strain are obtained by using the numerical Laplace transformation method with a MATLAB program. The results are discussed in the Sect. 4.

Results and discussions
In our strain gradient viscoelasticity theory, the gradient parameter is time-dependent and has a limited value, as shown in Eq. (64). At t ¼ 0, the gradient parameter is cj t¼0 ¼ c 0 . For t ! 1, the gradient parameter is cj t!1 ¼ c 0 . To present the trend of the gradient parameter over time, the normalized gradient parameter c=c 0 along the dimensionless time t=j g is illustrated in Fig. 2. Referring to the research conclusions for many cross-scale problems, the high-order effect is often predominated at the micro-/nano-scale. Similarly, the high-order viscosity related to the evolution in the micro-/nano-structure should be more rate-sensitive than the classical viscosity. Therefore, the classical relaxation time should be much larger than the high-order relaxation time, that is s g [ j g . From Fig. 2, it can be found that the normalized gradient parameter is always smaller than one except at t ¼ 0, that is, the viscoelastic gradient parameter c t ð Þ is always smaller than the one in the classical elasticity theory. Therefore, this strain gradient viscoelasticity could be used to explain the softening phenomenon of material at the micro-nanoscale. Besides, the normalized gradient parameter tends to its minimum value near the initial stage, that is, t=j g is small. The minimum value of 0.5 comes from the assumption of G 0 ¼ G 1 when plotting. The different G 0 and G 1 can characterized different materials that have different viscosity effects. When the value is fixed, the minimum value is c t =c 0 smaller.
In the mode-III crack problem, description about the stress near the crack tip are critical. Relations between the normalized stress near the crack tip and Fig. 2 The trend of the normalized gradient parameter over dimensionless time. The modulus G 0 ; G 1 are taken as the same value. The assumption of G 0 ¼ G 1 results in the minimum value is 0.5 time are shown in Fig. 3. From Fig. 3, it can be found that the normalized stress first increases and then decreases with time, and finally stabilizes. The gradient elasticity solution is related to the viscoelastic models and the corresponding modulus. When the classical relaxation time is much larger than the highorder relaxation time, that is, j g =s g is smaller, the normalized stress will take more time to stabilize, and the maximum stress is more significant. The stress tends to the gradient elasticity solution t ! 1, which is the steady-state value. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden.
The dimensionless stress over different distances from the crack tip is shown in Fig. 4. It can be found that the singularity of stress still exists and is negative. Meanwhile, for the gradient viscoelasticity solution, the dimensionless stress shows a peak after a distance away from the crack tip. The viscoelastic gradient stress is larger than the steady-state one which can be seen as the solution of classical gradient elasticity.
Obviously, the viscous gradient effect makes the stress field at the crack tip harden. This difference can also be explained by Eq. (64) or Fig. 2. Because the gradient parameter at t ¼ j g is smaller than the one at t ! 1, which is the gradient elasticity solution, the material is 'softer' at t ¼ j g and the stress will become larger. The position of the maximum stress will get smaller. The gradient parameter also affects the stress distribution near the crack tip. The stresses at the crack tip will be more significant for smaller length scale parameters.
The normalized stress near the crack tip as a function of the dimensionless time t j g with different distances from the crack tip is shown in Fig. 5a. It can be found that when the dimensionless distance is small, the maximum value of stress would increase with the increasing dimensionless distance. However, when x=l, especially, is over 0.1, the stress value of stress would decrease with the increasing of x=l, shown in Fig. 5b. For larger x=l, the normalized stress will take less time to stabilize. The stress tends to the gradient elasticity solution at t ! 1 the steady-state value, which is shown in Fig. 5c.

Conclusions
This paper presents the strain gradient viscoelastic analysis of mode-III crack without propagation in an infinite quasi-brittle advanced material using the strain gradient viscoelasticity theory and the Wiener-Hopf method with the time-dependent gradient parameter. During the derivation, the Wiener-Hopf method and the Laplace transformation are used to solve this viscoelastic gradient problem. Meanwhile, the Fig. 3 The normalized stress near the crack tip as a function of the dimensionless time t j g , with different j g =s g and G 1 =G 0 ¼ 1. The black dashed line denotes the normalized stress in the gradient elasticity solution.j g remain the same, x=l ¼ 1 G 0 ¼ G 1 when plotting. a 0 t=j g 1000. b 0 t=j g 10 solution to the gradient-dependent viscoelastic crack problem is obtained directly using the correspondence principle between the strain gradient viscoelastic theory and strain gradient elasticity theory. This viscoelastic gradient solution extends the gradient elastic ones presented by previously published studies . The viscous and size effects of the strain near the crack tip are analyzed. The results show that the stress increases as time increases and tends to a steady-state gradient elastic value finally, even if the gradient parameters are different.
Furthermore, stress components near the crack tip are discussed. Our viscoelastic gradient solution still has stress singularities, and the stress near the crack tip is time-dependent and related to the gradient parameter. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. As the distance from the crack tip decreases, the hardening effect becomes more significant, which is consistent with the 'researchers' understanding of the size effect. But when the distance is minimal, the hardening effect is no more significant. Fig. 4 The normalized stress near the crack tip as a function x=l at different moments, with different length scale parameters. a j g =s g ¼ 0:000001 G 1 =G 0 ¼ 0:000001 when plotting. b j g =s g ¼ 0:001 G 1 =G 0 ¼ 0:000001 when plotting. c j g =s g ¼ 0:001,G 1 =G 0 ¼ 0:001 when plotting. d j g =s g ¼ 0:000001, G 1 =G 0 ¼ 0:001 when plotting When the viscous effect is neglected or at t ! 1, the stress tends to the theoretical solution of strain gradient elasticity. that the order of authors listed in the manuscript has been approved by all of us. We understand that the Corresponding Author is the sole contact for the Editorial process. He/she is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs.
Author contributions K.D., Z.L. and Y.W. conceived and designed the project. K.D. and Y.W drafted the article and revised it critically. All authors reviewed the manuscript.
Funding This work was supported by the National Natural Science Foundation of China with grant nos. 11890681, 12032001, 11521202, and 11672301.

Declarations
Competing interests The authors declare no competing interests.
Competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 5
The normalized stress near the crack tip as a function of the dimensionless time t j g , with different distances from the crack tip. j g =s g is taken as 0.001, and G 1 =G 0 is taken as 0.001 when plotting. a x=l ! 0:3; 0:0001 t=j g 10. b x=l 0:1; 0:0001 t=j g 10. c x=l ! 0:3; 1000 t=j g 6000