# Peridynamic model for visco-hyperelastic material deformation in different strain rates

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## Abstract

This study presents a peridynamic (PD) constitutive model for visco-hyperelastic materials under homogenous deformation. The constitutive visco-hyperelastic model is developed in terms of Yeoh strain energy density function and Prony series. The material parameters in the model are identified by optimizing the classical stress–strain relation and tension test data for different strain rates. The peridynamic visco-hyperelastic force density function is proposed in terms of the peridynamic integral and the Yeoh strain energy density. The time-dependent behaviour for different strain rates is captured by numerical time integration representing the material parameters. The explicit form of peridynamic equation of motion is then constructed to analyse the deformation of visco-hyperelastic membranes. The numerical results concern the deformation and damage prediction for a polyurea membrane and membrane-type acoustic metamaterial with inclusions under homogenous loading. Different surface defects are considered in the simulation. The peridynamic predictions are verified by comparing with finite element analysis results.

## Keywords

Peridynamics Visco-hyperelastic materials Acoustic metamaterials Damage## 1 Introduction

Visco-hyperelastic materials have wide applications in aerospace, architecture, bioengineering and marine science, especially when they are loaded under large strain and wide range of strain rate. Metamaterials [1, 2] can also show such material characteristic. Visco-hyperelastic materials have significant dissipation properties such as instant bearings and high internal dampings. These dissipation properties lead these materials to undergo large deformations in relatively small stress and retain initial configuration after stress removal [3]. Based on these characteristics, visco-hyperelastic materials successfully reduce the vibration amplitude and minimize the rebound as shock absorber during large deformations [4].

The visco-hyperelastic materials consist of randomly oriented molecular chains which are cross-linked, spiral and tangled among themselves [5]. This configuration exhibits remarkable nonlinear mechanical behaviour including time dependency or stress softening phenomenon [6].

Constitutive models for visco-hyperelastic materials have been recently developed and can be divided into quasi-static and time-dependent models [7].

For quasi-static cases, the visco-hyperelastic materials exhibit hyperelastic behaviour. Hyperelastic behaviour is represented by empirical continuum approaches [8, 9]. These approaches require a vast amount of material parameters which usually leads to several deformation tests implemented for parameter identification. The physical interpretation of these parameters is often unclear [3]. On the other hand, some models are based on statistics of molecular chains network which are also been proposed with idealized assumptions [10, 11]. Moreover, Eremeyev and Naumenko [12] developed a relationship between effective work adhesion and peel force for thin hyperelastic films undergoing large deformation.

Time-dependent models associated with viscoelasticity phenomenon have been mostly formulated by hereditary integral approaches or experimental methods. In these models, the effect of strain history on the current stress state is reflected by fading-memory function by considering different strain rates [13, 14, 15]. Some other models are based on the constitutive relation between strain energy density and strain invariants to include the effect of time-dependent behaviour. The choice for appropriate strain energy function is determined by material parameters identification [3]. By using these models, the strain rate dependency can be considered to predict the large deformation behaviour of visco-hyperelastic materials [16].

However, the classical continuum mechanics (CCM) for visco-hyperelastic materials faces mathematical challenges in the presence of defects. This is due to the fact that the equation of motion in CCM, including well-known finite element method [17, 18], involves spatial derivatives for displacement components. In addition to CCM, there exist variational approaches with and without considering damage [19, 20, 21]. As an alternative approach, the peridynamic theory, a new continuum mechanics formulation introduced by Silling [22, 23], removes the aforementioned challenges of CCM using the integral representation of internal forces. Therefore, it is suitable for damage prediction in structures. According to dell’Isola et al. [24, 25], the origins of peridynamics go back to Gabrio Piola, who has made many contributions in mathematics and physics, especially nonlocal continuum theory, including nonlocal internal interactions, strong form of variational principle and higher gradient mechanics. Besides, PD theory is not limited to linear elastic material [26]. Madenci [27] provided the PD integrals for strain invariants to investigate the PD form of the strain energy density for both linear elastic and hyperelastic materials. It was concluded that the deformation response based on new strain energy density function is similar to that of bond-based peridynamics. Bang and Madenci [28] developed the PD strain energy density form for neo-Hookean-type hyperelastic membrane under different loading conditions. Madenci and Oterkus [29] proposed the ordinary state-based PD constitutive model for viscoelastic deformation in terms of Prony series. They captured the relaxation behaviour of viscoelastic material under mechanical and thermal loads. Madenci and Oterkus [30] and Mitchell [31] presented the ordinary state-based PD model for plastic deformation. Taylor [32] and Forster et al. [33] also considered the PD model for viscoplastic materials.

In this study, PD model for visco-hyperelastic materials is developed. This study first presents a Yeoh-type visco-hyperelastic constitutive model which works in a broad range of strain rates. The hyperelastic part of Cauchy stress captures the quasi-static behaviour using the Yeoh strain energy density, while the viscoelastic part of Cauchy stress represents the rate-dependent behaviour by using hereditary integral. This constitutive model is verified by experimental data for high-damping rubber [34] and polyurea [35]. Secondly, based on PD integrals for the first strain invariant of right Cauchy–Green strain tensor [27], the derivation of PD force density is presented by using Yeoh visco-hyperelastic material parameters and Cauchy stress definition. In order to capture the time-dependent properties of viscoelastic materials, the PD viscoelastic force density is described in terms of Prony series. The developed PD model is verified by comparing with finite element predictions. The developed PD model is used to simulate polyurea membrane with a defect in the form of a hole and a pre-existing crack. Finally, the PD model is used to predict membrane-type acoustic metamaterial morphology.

## 2 Visco-hyperelastic constitutive model

### 2.1 Hyperelasticity

*W*represents the strain energy density function which can be defined by the invariants of \(\mathbf{B}\).

*nd*represents the number of dimensions and \(c_{n0} \) represents material parameters. The 3-D Yeoh model can be written as

### 2.2 Visco-hyperelasticity

*t*represents time and \(\tau \) represents time integral variable.

*t*can be expressed as \(\lambda =\dot{\varepsilon }_{11} t+1\). An infinitesimal deformation variable \(\zeta \) which satisfies \(\zeta =\dot{\varepsilon }_{11} \tau +1\) where \(1\le \zeta \le \lambda \) is introduced to substitute the integral variable \(\tau \) in Eq. (28). Therefore, the relation between \(\lambda \) and \(\tau \) becomes:

### 2.3 Identification of constitutive model parameters

Predicted visco-hyperelastic parameters from experimental data by Amin et al. [34]

Hyperelastic parameters | Viscoelastic parameters | |||||
---|---|---|---|---|---|---|

Material parameters | \(c_{10}\) (MPa) | \(c_{20}\) (MPa) | \(\tilde{c}^{1}_{10}\) (MPa) | \(\tilde{c}^{1}_{20}\) (MPa) | \(\varphi _\mathrm{ref}\) | \(\eta \) |

Value | 0.1438 | 0.1438 | 0.0587 | 0.0562 | 0.8673 | 0.1 |

Figure 2 represents the comparison of predicted and experimental stretch and stress relations in different strain rates for HDR material. As it can be seen from the figure, there is a good agreement between the proposed visco-hyperelastic model results and experimental data. Amin et al. [34] reported that the behaviour of HDR did not show significant change for strain rates higher than \(0.88\,{\hbox {s}}^{-1}\).

Predicted visco-hyperelastic parameters from experimental data by Roland et al. [35]

Hyperelastic parameters | Viscoelastic parameters | |||||
---|---|---|---|---|---|---|

Material parameters | \(c_{10} \)(MPa) | \(c_{20} \)(MPa) | \(\tilde{c}^{1}_{10} \) (MPa) | \(\tilde{c}^{1}_{20} \)(MPa) | \(\varphi _\mathrm{ref} \) | \(\eta \) |

Value | 0.1996 | 0.0170 | 0.01733 | 0.0015 | 0.9466 | 1.6665 |

Figure 3 represents the comparisons of predicted and experimental stretch and stress relations in different strain rates for polyurea material. As it can be seen from the figure, there is a good agreement between the proposed model results and experimental data even in higher strain rates, illustrating the applicability of the proposed model for the wide broadband of strain rates.

## 3 Peridynamic theory

Peridynamic theory concerns the nonlocal representation of a physical field at a material point. Each material point has nonlocal interaction with other material points in a certain horizon, and the strength of interaction is specified by a weight function.

*t*can be defined as [39]

According to the PD theory, the displacement field at material point \(\mathbf{x}\) is influenced by the collective interaction of all the family members \(\mathbf{{x}'}\) located in its interaction domain, \(H_\mathbf{x} \), which is defined by its horizon, \(\delta \), as shown in Fig. 4 [22].

*N*denotes the number of family members of material point \(\mathbf{x}_{(k)}\) and \(V_{(j)} \) represents the volume of each material point \(\mathbf{x}_{(j)} \).The parameters \(\mathbf{t}_{(k)(j)} \) and \(\mathbf{t}_{(j)(k)} \) are the force density vectors being parallel to the relative position vector in the deformed state as shown in Fig. 4. As derived by Madenci and Oterkus [40], determination of force density vectors requires the explicit PD form of strain energy density function. If the PD form of the strain energy density function is known, the PD force density vector can be obtained as

*m*in Eq. (37) is defined as

*m*can be evaluated as \(m=V\delta ^{2}\) with

*V*representing the volume as \(V=4\pi \delta ^{3}/3\) for 3D, \(V=\pi h\delta ^{2}\) for 2D and \(V=2A\delta ^{2}\) for 1D [27]. The parameter

*h*denotes the thickness of the membrane, and

*A*denotes the cross-sectional area of the bar.

*m*is specified as \(m=4\pi \delta ^{5}/3\) for 3-D, \(m=\pi h\delta ^{4}\) for 2-D and \(m=2A\delta ^{4}\) for 1-D models.

## 4 Visco-hyperelastic Peridynamic formula

### 4.1 Hyperelastic response based on Yeoh model

### 4.2 Visco-hyperelastic response based on Yeoh model

*i*represents the terms of Prony series and the parameter \(\theta _i \) represents the relaxation time of

*i*th Prony series. \(W_i^v \) is the Yeoh model for viscous strain energy density given in Eq. (17).

### 4.3 Failure criteria

*h*denoting the thickness as [29, 30]

## 5 Numerical procedure

- (1)At the load step \(\left[ {n+1} \right] \), the viscous stretch between material points \(\mathbf{x}_{(k)}\) and \(\mathbf{x}_{(j)}\) can be calculated by using Eq. (60) asin which \(\Delta \lambda _\mathrm{ins}^{(k)(j)} =\lambda _\mathrm{ins}^{(k)(j)} \left( {t_{n+1} } \right) \,-\lambda _\mathrm{ins}^{(k)(j)} \left( {t_n } \right) \,\), where \(\lambda _\mathrm{ins}^{(k)(j)} =\frac{\left| {\mathbf{y}_{(j)} -\mathbf{y}_{(k)} } \right| }{\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| }\) and \(\theta =\varphi _\mathrm{ref} \cdot \bar{{\dot{\varvec{\upvarepsilon }}}}^{-\eta } \)$$\begin{aligned} \gamma _{(k)(j)} \left( {t_{n+1} } \right) \,= & {} \left( {\gamma _{(k)(j)} \left( {t_n } \right) +\Delta \lambda _\mathrm{ins}^{(k)(j)} } \right) -\left[ {\gamma _{(k)(j)} \left( {t_n } \right) +\frac{1}{2}\Delta \lambda _\mathrm{ins}^{(k)(j)} } \right] \nonumber \\&\cdot \left( {\frac{\Delta t}{\theta }} \right) +\frac{1}{2}\cdot \gamma _{(k)(j)} \left( {t_n } \right) \cdot \left( {\frac{\Delta t}{\theta }} \right) ^{2} \end{aligned}$$(70)
- (2)By substituting Eq. (70) into Eq. (55), the time integral for the load step \(\left[ {n+1} \right] \) can be calculated asin which \(\tilde{\mathrm{c}}_{10}^1\) and \(\tilde{\mathrm{c}}_{20}^1\) represent material parameters for viscous property and \(I_1 \) can be calculated for homogenous deformation loading \(\lambda \) by using Eq. (52).$$\begin{aligned} J_{(k)(j)} \left( {t_{n+1} } \right) =2\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| \tilde{c}_{10}^1 \gamma _{(k)(j)} \left( {t_{n+1} } \right) +4\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| \tilde{c}_{20}^1 \left( {I_1 -2} \right) \gamma _{(k)(j)} \left( {t_{n+1} } \right) \end{aligned}$$(71)
- (3)Calculate the weight function between material points \(\mathbf{x}_{(k)}\) and \(\mathbf{x}_{(j)}\) by using Eq. (38) as:$$\begin{aligned} w\left( {\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| } \right) =\left( {\frac{\delta }{\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| }} \right) ^{2} \end{aligned}$$(72)
- (4)Calculate the viscous force density between material points \(\mathbf{x}_{(k)} \) and \(\mathbf{x}_{(j)} \) for the load step \(\left[ {n+1} \right] \) by using Eqs. (51), (71) and (72) asin which \(m=\pi h\delta ^{4}\) for 2-D [27].$$\begin{aligned} \mathbf{t}_{(k)(j)}^v \left( {t_{n+1} } \right) =\frac{2}{m}w\left( {\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| } \right) J_{(k)(j)} \left( {t_{n+1} } \right) \frac{\mathbf{y}_{(j)} \mathbf{-y}_{(k)} }{\left| {\mathbf{y}_{(j)} \mathbf{-y}_{(k)} } \right| } \end{aligned}$$(73)
- (5)The corresponding hyperelastic force density between material points \(\mathbf{x}_{(k)} \) and \(\mathbf{x}_{(j)}\) can be calculated from Eq. (47) asin which \(I_1 \) can be calculated for homogenous deformation loading \(\lambda \) by using Eq. (52) and \(c_{10}\), \(c_{20}\) represent material parameters for hyperelastic property.$$\begin{aligned} \mathbf{t}_{(k)(j)}^h \left( {t_{n+1} } \right) =\,\frac{2}{\pi h\delta ^{4}}\left[ {\left( {c_{10} +2c_{20} (I_1 -2)} \right) } \right] \left[ {2w\left( {\left| {\mathbf{x}_{(j)} -\mathbf{x}_{(k)} } \right| } \right) \left| {\mathbf{y}_{(j)} \mathbf{-y}_{(k)} } \right| } \right] \frac{\mathbf{y}_{(j)} \mathbf{-y}_{(k)} }{\left| {\mathbf{y}_{(j)} \mathbf{-y}_{(k)} } \right| } \end{aligned}$$(74)
- (6)The total force density between material points \(\mathbf{x}_{(k)} \) and \(\mathbf{x}_{(j)} \) for the load step \(\left[ {n+1} \right] \) can be calculated as$$\begin{aligned} \mathbf{t}_{(k)(j)} \left( {t_{n+1} } \right) =\mathbf{t}_{(k)(j)}^h \left( {t_{n+1} } \right) +\mathbf{t}_{(k)(j)}^v \left( {t_{n+1} } \right) \end{aligned}$$(75)
- (7)The micropotential \(w_{\left( k \right) \left( j \right) } \) for the load step \(\left[ {n+1} \right] \) between two material points \(\mathbf{x}_{\left( k \right) }\) and \(\mathbf{x}_{\left( j \right) }\) can be calculated by using the force density from Eq. (64) as$$\begin{aligned} w_{\left( k \right) \left( j \right) } \left( {t_{n+1} } \right) =\int \limits _0^{s_{\left( k \right) \left( j \right) } (t=0^{+})} {t_{0^{+}\left( k \right) \left( j \right) } \left| {\mathbf{x}_{\left( k \right) } -\mathbf{x}_{\left( j \right) } } \right| } ds_{\left( k \right) \left( j \right) } +\int \limits _{s_{\left( k \right) \left( j \right) } (t=0^{+})}^{s_{\left( k \right) \left( j \right) } (t)} {\mathbf{t}_{(k)(j)} \left( {t_{n+1} } \right) \left| {\mathbf{x}_{\left( k \right) } -\mathbf{x}_{\left( j \right) } } \right| } \mathrm{d}s_{\left( k \right) \left( j \right) }\nonumber \\ \end{aligned}$$(76)
- (8)The energy release rate of the interaction between two material points \(\mathbf{x}_{\left( k \right) }\) and \(\mathbf{x}_{\left( j \right) } \) for the load step \(\left[ {n+1} \right] \) can be calculated from Eq. (66) asleading to the failure parameter for the interaction between two material points \(\mathbf{x}_{\left( k \right) }\) and \(\mathbf{x}_{\left( j \right) }\) by using Eq. (68), and local damage for the material point \(\mathbf{x}_{\left( k \right) } \) by using Eq. (69).$$\begin{aligned} G_{\left( k \right) \left( j \right) } \left( {t_{n+1} } \right) =\frac{1}{h\Delta x}w_{\left( k \right) \left( j \right) } \left( {t_{n+1} } \right) V_{\left( k \right) } V_{\left( j \right) } \end{aligned}$$(77)

## 6 Numerical results

Secondly, the visco-hyperelastic membrane under situations of without defect, with a defect in the form of a hole and a pre-existing crack and MAM-type structure are validated. The visco-hyperelastic parameters based on Yeoh model and Prony series, \(c_{10} \), \(c_{20} \), \(\tilde{c}^{1}_{10} \), \(\tilde{c}^{1}_{20}\), \(\varphi _\mathrm{ref} \) and \(\eta \), are given in Table 2.

The membranes are discretized into uniform grids. The discretization is chosen as \(100\times 100\) for the membrane without defect and with a defect of a hole and a pre-existing crack. In terms of the membrane-type acoustic material (MAM), the discretization is chosen as \(120\times 120\) in order to capture the interaction between multiple materials. In view of the convergence for PD predictions, the horizon in this study is specified as \(\delta =3.015\Delta x\).

The displacement boundary condition is imposed through the boundary layers. The crack propagation problems are investigated by using dynamic analysis with a time step size of \(\Delta t=10^{\mathrm{-4}}\,\hbox {s}\). All other cases are investigated for quasi-static analysis by using ADR method.

In order to verify the accuracy of PD model, the membrane is also simulated by FEM. The acceptable finite element discretization is achieved by \(100\times 100\). The element type PLANE 182 with Yeoh model and Prony series are utilized in the ANSYS model.

### 6.1 Hyperelastic membrane without a defect

It is obvious that the PD stress result for the hyperelastic membrane is greater than the result for the elastic membrane. As can be seen from Fig. 7, the difference between PD stress is \(0.31\times 10^{6}\) Pa for \(\lambda =1.5\) and \(9.12\times 10^{6}\) Pa for \(\lambda =4\). The increasing deviation between the PD stress predictions indicates the hyperelastic material property of the membrane.

### 6.2 Hyperelastic membrane with a hole

Figures 9, 10 represent the displacement components for \(\lambda =4\) and \(\lambda =7\) in the deformed configuration. As it can be seen from the figures, the shape of the hole remains circular. The contour plots of PD deformation predictions agree well with the ANSYS predictions, validating the PD model even for large deformations.

### 6.3 Hyperelastic membrane with a pre-existing crack

In the presence of a pre-existing crack, the hyperelastic membrane in Fig. 11 includes a vertical crack with a length of \(l=0.1 \hbox {m}\) in the centreline of the membrane.

In the situation not allowing failure, the membrane is subjected to homogeneous principle stretch as \(\lambda =1.5\) and \(\lambda =4\). Figures 12, 13 represent the displacement components in the deformed configuration. As it can be seen from the results, PD predictions compare well with ANSYS results. There is a slight difference near the crack tips between the PD and ANSYS results. This is due to more refined mesh in ANSYS model near the crack tips. In Fig. 13, as expected, the crack opening increases with increased loading.

*x*- and

*y*-directions simultaneously. As the initial crack is aligned with

*y*-direction, the top and bottom part of failure zone continuously expands as crack branches.

### 6.4 Hyperelastic membrane with rigid inclusions

The membrane-type acoustic metamaterial (MAM) is generally constructed by elastic membrane and rigid inclusions. The solid inclusions are bonded into the membrane. As shown in Fig. 15a, four inclusions and a hyperelastic membrane are considered to construct the MAM. The radius of solid inclusions is \(r_0 =0.0625\), and their centres are located at \(\left( {-2.5r_0 ,0} \right) \), \(\left( {2.5r_0 ,0} \right) \), \(\left( {0, -1.5r_0 } \right) \) and \(\left( {0,1.5r_0 } \right) \). Since the hyperelastic membrane has a Young’s modulus of \(E=\hbox {1.1976 MPa}\), the material parameters for solid inclusions are chosen as \(E_2 =144\,\hbox {GPa}\) [46] and \(\nu =0.3\).

Note that if the material point is inside the rigid inclusion, the force density is calculated based on Eq. (79); on the other hand, if the material point is inside the membrane, the force density is calculated based on Eq. (74).

In the absence of a pre-existing crack, the MAM is subjected to homogenous principle stretch as \(\lambda =1.5\). Figure 16 represents the displacement components in the deformed configuration. As can be seen from this figure, the PD predictions are in good agreement with those of ANSYS. The predictions capture the effect interaction of the solid inclusions with the hyperelastic membrane.

In the presence of a crack, MAM includes a horizontal crack with length of \(l=0.1 \hbox {m}\) as shown in Fig. 15b. The PD predictions of displacement contour plots for \(\lambda =1.5\) are presented in Fig. 17. Comparing with displacement contours of PD and ANSYS results, the effect of surrounding inclusions is accurately captured.

When failure is allowed, the critical energy release rate is chosen as \(G_\mathrm{c} =6.69\,N/m\) for hyperelastic-type membrane. The homogenous loading stretch is specified as \(\dot{\lambda }=1 \left( {1/\upmu \hbox {s}} \right) \) with \(dt=0.1\,\upmu \hbox {s}\). Figure 18 represents the crack propagation at the end of \(\hbox {150 } \upmu \hbox {s}\), \(\hbox {170 } \upmu \hbox {s}\), \(\hbox {180 } \upmu \hbox {s}\) and \(\hbox {190 } \mu \hbox {s}\). The maximum damage values are observed, especially close to the tips of initial crack. The failure zone expands in both *x*- and *y*-directions. The crack propagates around the edge of the solid inclusions since the critical energy release rate for solid inclusions is much greater than for the hyperelastic membrane.

### 6.5 Visco-hyperelastic membrane without a defect

In the absence of defect, the visco-hyperelastic membrane is subjected to the homogenous principle stretch \(\lambda \) which is varied from 1.25 to 2.5 with an increment of 0.25. The visco-hyperelastic behaviour is evaluated at \({t}=2.5\,\upmu \hbox {s}\), and strain rate is specified as \(\dot{\varepsilon }_{11} =0.15\).

Figure19 presents the PD stress at \(\left( {x=-0.165\,\,\hbox {m},y=0.125\,\, \hbox {m}} \right) \) for both visco-hyperelastic and hyperelastic membrane. As it can be seen from Fig. 19, PD stresses for hyperelastic membrane are bigger than those of visco-hyperelastic membrane. The difference between PD stress is \(0.24\times 10^{5}\) Pa for \(\lambda =1.25\) and \(1.20\times 10^{6}\) Pa for \(\lambda =2.5\). The stiffness property of visco-hyperelastic membrane degrades with time, leading to the increasing deviation between PD stress of hyperelastic and visco-hyperelastic membranes.

### 6.6 Visco-hyperelastic membrane with a hole

In the presence of a hole, the visco-hyperelastic membrane is subjected to the applied loading stretch of \(\lambda =2.5\) and \(\lambda =3.0\). The constant strain rate is specified as \(\dot{\varepsilon }_{11} =0.15\).

Figures 20, 21 present the displacement components in the deformed configuration. As it can be seen from the figures, the shape of the hole remains circular. The contour plots of PD deformation predictions agree well with the ANSYS predictions.

### 6.7 Visco-hyperelastic membrane with a pre-existing crack

In the presence of pre-existing crack, the visco-hyperelastic membrane is subjected to homogeneous principle stretch as \(\lambda =1.5\) and \(\lambda =2.5\).

Figures 22, 23 present the displacement components in the deformed configuration. The good agreement between PD deformation contour plots and those of ANSYS validates the PD model for visco-hyperelastic material.

*y*-direction, the crack initially propagates in

*y*-direction. Later, the crack propagates in both

*x*- and

*y*-directions as it can be seen in Fig. 24c, d. The damage pattern is also presented in deformed configuration in Fig. 25. It is observed that a blunted zone occurs at the tips of the crack as shown in Fig. 25.

### 6.8 Visco-hyperelastic membrane with rigid inclusions

## 7 Conclusion

The novelty of this study is to develop a PD model for visco-hyperelastic material. It specially concerns Yeoh-type membrane under homogenous stretch loading conditions. The Yeoh strain energy density is expressed by the invariants of Cauchy–Green tensor. The material parameters in the strain energy function are identified by curve fitting of tension test data. The PD form of the visco-hyperelastic force density function is represented in terms of the PD integral. The viscous stretch term for viscoelastic properties is derived based on Yeoh strain energy density. The developed PD model is utilized to predict the deformation and damage for a polyurea membrane with a defect in the form of a hole and a pre-existing crack. The deformation and damage for a membrane-type acoustic metamaterial with inclusions are also simulated by using the developed PD model. The numerical results capture the quasi-static deformation and dynamic fracture propagation of the polyurea membrane and membrane-type acoustic metamaterial under homogenous loading. The developed PD model is verified by obtaining good agreement with ANSYS predictions.

## Notes

### Acknowledgements

This work was financially supported by the Shaanxi Province Industrial Science and Technology Project (Grant No. 2016GY-111), the National Natural Science Foundation of China (Grants Nos. 11474230 and 11704314), the National Key Research and Development Program of China (Grant No. 2016YFF0200902), the International Cooperation Training Program for the Innovative Talents of China Scholarship Council and University of Strathclyde.

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