Modified bond-based peridynamic approach for modeling the thermoviscoelastic response of bimaterials with viscoelastic–elastic interface

Abstract

This study investigates the constitutive relationships for a modified bond-based peridynamic (MBB-PD) model to analyze bimaterial structures with a viscoelastic component subjected to mechanical and thermal loads. The hereditary integral viscoelastic constitutive relationship is used to model the viscoelastic behavior of a generalized Maxwell model, with viscoelastic constitutive coefficients obtained from the Prony series. To model the bimaterial interface between viscoelastic and elastic segments, various functions are proposed. Additionally, normal and shear bonds are defined as failure criteria based on critical stretches and angles. The fidelity of the proposed model is evaluated by comparing its thermoviscoelastic creep and recovery responses with finite element (FE) solutions for single and bimaterial plates. The sensitivity of the interface function to the horizon size is also examined. The results demonstrate that small horizon sizes still yield satisfactory results near the interface while reducing computational costs. Finally, the proposed MBB-PD model is applied to simulate a double cantilever beam (DCB) delamination test with a rubber interface, with predictions that are in good agreement with those from the extended finite element method (XFEM). The MBB-PD model proves to be a useful and efficient tool for analyzing bimaterial structures with viscoelastic components.

Appendices

Appendix A: Modified bond-based true strain tensor

The MBB-PD model employs imaginary local elements for every bond to determine all deformation components and their corresponding true strain components. As displayed in Fig. 

Fig. 14

The local element and participating points in the undeformed body and deformed body due to a and b EHNSB, c and d EVNSB, e and f ESSB

14, the interaction between $j$ and $k$ in the undeformed configuration can be an EHNSB, EVNSB, or ESSB. The imaginary local rectangular elements are utilized to capture the transformation of these bonds during deformation.

As the deformed local element of the EHNSB, illustrated in Fig. 14b, assuming infinitesimal strain, the EHNSB is associated with normal, transverse, and shear true strains that are estimated as:

$${\varepsilon }_{11 }^{x}\approx \frac{{u}_{1\left(j\right)}-{u}_{1\left(k\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(k\right)}}$$

$$\begin{array}{c}{\varepsilon }_{22}^{x}\approx \frac{1}{2}\left(\frac{{u}_{2\left(a\right)}-{u}_{2\left(b\right)}}{{y}_{2\left(a\right)}-{y}_{2\left(b\right)}}+\frac{{u}_{2\left(c\right)}-{u}_{2\left(d\right)}}{{y}_{2\left(c\right)}-{y}_{2\left(d\right)}}\right)\end{array}$$

$${\varepsilon }_{12}^{x}\approx \frac{1}{2}\left\{\left(\frac{{u}_{1\left(a\right)}-{u}_{1\left(b\right)}}{{y}_{2\left(a\right)}-{y}_{2\left(b\right)}}+\frac{{u}_{2\left(j\right)}-{u}_{2\left(k\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(k\right)}}\right)+\left(\frac{{u}_{1\left(c\right)}-{u}_{1\left(d\right)}}{{y}_{2\left(c\right)}-{y}_{2\left(d\right)}}+\frac{{u}_{2\left(j\right)}-{u}_{2\left(k\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(k\right)}}\right)\right\}$$

(38)

where ${\varepsilon }_{\alpha }^{x}, \alpha \in \left\{\mathrm{11,22,12}\right\}$ indicate the normal axial, transverse, and shear true strain components of an EHNSB. Additionally, the variables ${u}_{\alpha \left(\beta \right)}$ and ${y}_{\alpha \left(\beta \right)}$ represent the $\alpha$ components of the displacement and deformed position of the material point $\beta$, respectively.

Similar to the EHNSB, the EVNSB’s true strain components, based on the deformed local element of the EVNSB (Fig. 14d), can be approximated as follows:

$${\varepsilon }_{11}^{y}\approx \frac{1}{2}\left(\frac{{u}_{1\left(a\right)}-{u}_{1\left(b\right)}}{{y}_{1\left(a\right)}-{y}_{1\left(b\right)}}+\frac{{u}_{1\left(c\right)}-{u}_{1\left(d\right)}}{{y}_{1\left(c\right)}-{y}_{1\left(d\right)}}\right)$$

$${\varepsilon }_{22}^{y}\approx \frac{{u}_{2\left(j\right)}-{u}_{2\left(k\right)}}{{y}_{2\left(j\right)}-{y}_{2\left(k\right)}} $$

$${\varepsilon }_{12}^{y}\approx \frac{1}{2}\left\{\left(\frac{{u}_{1\left(j\right)}-{u}_{1\left(k\right)}}{{y}_{2\left(j\right)}-{y}_{2\left(k\right)}}+\frac{{u}_{2\left(a\right)}-{u}_{2\left(b\right)}}{{y}_{1\left(c\right)}-{y}_{1\left(d\right)}}\right)+\left(\frac{{u}_{1\left(j\right)}-{u}_{1\left(k\right)}}{{y}_{2\left(j\right)}-{y}_{2\left(k\right)}}+\frac{{u}_{2\left(c\right)}-{u}_{2\left(d\right)}}{{y}_{1\left(c\right)}-{y}_{1\left(d\right)}}\right)\right\}$$

(39)

where ${\varepsilon }_{\alpha }^{y}, \alpha \in \left\{\mathrm{11,22,12}\right\}$ are the normal axial, transverse, and shear true strain components of an EVNSB.

The deformed local element of ESSB is indicated in Fig. 14f. Thus, the true strains of ESSB can be calculated as follows:

$$\begin{array}{c}{\varepsilon }_{11}^{xy}\approx \frac{1}{2}\left(\frac{{u}_{1\left(j\right)}-{u}_{1\left(b\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(b\right)}}+\frac{{u}_{1\left(k\right)}-{u}_{1\left(a\right)}}{{y}_{1\left(k\right)}-{y}_{1\left(a\right)}}\right)\end{array}$$

$${\varepsilon }_{22}^{xy}\approx \frac{1}{2}\left(\frac{{u}_{2\left(j\right)}-{u}_{2\left(a\right)}}{{y}_{2\left(j\right)}-{y}_{2\left(a\right)}}+\frac{{u}_{2\left(k\right)}-{u}_{2\left(b\right)}}{{y}_{2\left(k\right)}-{y}_{2\left(b\right)}}\right)$$

$${\varepsilon }_{12}^{xy}\approx \frac{1}{2}\left\{\left.\begin{array}{c}\left(\frac{{u}_{1\left(j\right)}-{u}_{1\left(a\right)}}{{y}_{2\left(j\right)}-{y}_{2\left(a\right)}}+\frac{{u}_{2\left(j\right)}-{u}_{2\left(b\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(b\right)}}\right)+\left(\frac{{u}_{1\left(k\right)}-{u}_{1\left(b\right)}}{{y}_{2\left(k\right)}-{y}_{2\left(b\right)}}+\frac{{u}_{2\left(j\right)}-{u}_{2\left(a\right)}}{{y}_{1\left(j\right)}-{y}_{1\left(a\right)}}\right)\end{array}\right\}\right.$$

(40)

where ${\varepsilon }_{\alpha }^{xy}, \alpha \in \left\{11, 22, 12\right\}$ are the normal horizontal true strain, normal vertical true strain, and true shear strain of an ESSB.

Appendix B: Modified bond-based correction factors

A combined loading condition model obtains the components of the MBB-PD correction factor matrix in the CCM and PD frameworks. The process of calibration relies on comparing the strain energy densities that are computed from each framework. When the strain energies of PD and CCM are made equal, it produces the components of the correction factor matrix [45]:

$${\lambda }_{xx}=2{V}_{\left(j\right)}{\left(m\left(m+1\right)\Delta x+\sum_{k=1}^{{N}_{\left(j\right)}^{S}}\begin{array}{c}\left({\left(1+\frac{1}{2}\mathrm{ tan}{\theta }_{jk}\right)}^{2}\begin{array}{c}\left|\begin{array}{c}{\overrightarrow{\xi }}_{jk}\end{array}\right|\end{array}\right)\end{array}\right)}^{-1}$$

$${\lambda }_{yy}=2{{V}_{\left(j\right)}\left(m\left(m+1\right)\Delta y+\sum_{k=1}^{{N}_{\left(j\right)}^{S}}\begin{array}{c}\left({\left(1+\frac{1}{2}\mathrm{ cot}{\theta }_{jk}\right)}^{2}\begin{array}{c}\left|\begin{array}{c}{\overrightarrow{\xi }}_{jk}\end{array}\right|\end{array}\right)\end{array}\right)}^{-1}$$

$${\lambda }_{xy}=2{V}_{\left(j\right)}{\left(\sum_{k=1}^{{N}_{\left(j\right)}^{S}}\begin{array}{c}\left({\left(\mathit{csc}{2\theta }_{jk}+1\right)}^{2}\begin{array}{c}\left|\begin{array}{c}{\overrightarrow{\xi }}_{jk}\end{array}\right|\end{array}\right)\end{array}\right)}^{-1}$$

(41)

where $\Delta x$ and $\Delta y$ are the spacing between two adjacent particles in the horizontal and vertical directions, respectively, and $m=\frac{\delta }{\Delta x}$. With uniform discretization ($\Delta x=\Delta y=\Delta$) for a square symmetric region, they can be simplified as follows:

$${\lambda }_{xx}={\lambda }_{yy}=\frac{2}{m\left(m+1\right)V\Delta }$$

$${\lambda }_{xy}=\frac{2}{{m}^{2}{\left(m+1\right)}^{2}V\Delta }$$

(42)