*Article* **Deformation and Strength Parameters of a Composite Structure with a Thin Multilayer Ribbon-like Inclusion**

**Volodymyr Hutsaylyuk 1,\*, Yosyf Piskozub 2, Liubov Piskozub <sup>2</sup> and Heorhiy Sulym <sup>3</sup>**


**Abstract:** Within the framework of the concept of deformable solid mechanics, an analyticalnumerical method to the problem of determining the mechanical fields in the composite structures with interphase ribbon-like deformable multilayered inhomogeneities under combined force and dislocation loading has been proposed. Based on the general relations of linear elasticity theory, a mathematical model of thin multilayered inclusion of finite width is constructed. The possibility of nonperfect contact along a part of the interface between the inclusion and the matrix, and between the layers of inclusion where surface energy or sliding with dry friction occurs, is envisaged. Based on the application of the theory of functions of a complex variable and the jump function method, the stress-strain field in the vicinity of the inclusion during its interaction with the concentrated forces and screw dislocations was calculated. The values of generalized stress intensity factors for the asymptotics of stress-strain fields in the vicinity of the ends of thin inhomogeneities are calculated, using which the stress concentration and local strength of the structure can be calculated. Several effects have been identified which can be used in designing the structure of layers and operation modes of such composites. The proposed method has shown its effectiveness for solving a whole class of problems of deformation and fracture of bodies with thin deformable inclusions of finite length and can be used for mathematical modeling of the mechanical effects of thin FGM heterogeneities in composites.

**Keywords:** functionally gradient material; composite; thin inhomogeneity; fracture mechanics; nonperfect contact; stress intensity factor

#### **1. Introduction**

Microscopic, layered structures in fields such as microelectronics, biotechnology, energy, weaponry, etc. are gaining special attention in modern engineering and technology. Among the most important scientific projects, experts identify a significant increase in computer performance, restoration of human organs using reproduced tissues (obtained from 3D printers) and obtaining new structured materials created directly from given molecules and atoms. Quite often these inclusions are used as elements to reinforce structural parts of machines and structures or as fillers of composite materials. Thin lamellar inhomogeneities are also a characteristic phenomenon at the interphase boundaries of crystalline grains arising during crystallization [1–7]. In this regard, there is a need to provide mathematical modeling of nanostructure mechanics, which is still a pressing problem of materials science theory. At this stage of the development of mechanics, it is already possible to concentrate on the construction of the complex universal equations suitable for investigations of multiscale, including layered, structures and the development of methods for their solution.

**Citation:** Hutsaylyuk, V.; Piskozub, Y.; Piskozub, L.; Sulym, H. Deformation and Strength Parameters of a Composite Structure with a Thin Multilayer Ribbon-like Inclusion. *Materials* **2022**, *15*, 1435. https://doi.org/10.3390/ma15041435

Academic Editor: Andrea Spagnoli

Received: 20 January 2022 Accepted: 12 February 2022 Published: 15 February 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In such structures, each layer or their combination has its functional purpose, in particular, anti-corrosion, anti-abrasion, heat protection, strengthening to inhibit and block crack growth, reduce porosity, provide a high degree of adhesion of the components [8–14]. Thanks to multilayers, it is also possible to increase the service life of structures, and their use can significantly reduce material intensity and cost and increase the endurance of products. At the same time in a structure with thin layers, there is a concentration of stresses near places of change of physical and mechanical characteristics of materials. And it is the higher, the greater the difference in their properties.

Inhomogeneous structures with optimally varying physical and mechanical properties along with the thickness, known as functionally graded materials (FGMs) [15–20], allow one to reduce such stress concentrations in the vicinity of the contact between the matrix and the interlayer by avoiding abrupt transitions in the properties of the components. A detailed review of the manufacturing techniques can be found in [21–26]. FGMs are often used in the coatings of structural elements to protect them from the harmful effects of temperature [8–11,27–35], etc. One of the frequently used variants of FGM arrangement is the combination of ceramics with metal [36,37], but this often leads to the violation of the contact between them. Due to the brittle nature of ceramics, there is a need for additional research into the applicability limits of such FGM structures [38–40]. The complexity of the geometry of structural elements and consideration of imperfections in the contact of their components stimulate the process of improving mathematical models of FGMs to ensure their qualitative design both in terms of mechanical strength [12–14,41–54] and in terms of consideration of thermal, magnetic, piezoelectric loading factors [47,48]. The use of the FGMs seems to be one of the most effective materials in the realization of sustainable development in industries.

An important aspect of strength research, including tensile strength, for such structures, is to improve their strength criteria, to determine such key parameters as stress intensity factors (SIF) in the points of singularity. Moreover, since we consider thin inhomogeneities not only in the form of classical cracks but also thin cavities filled with an arbitrary elastic or nonlinearly elastic material, it makes sense to claim that the theory of thin inclusions is an essential generalization of the crack theory and the so-called generalized SIFs, which characterize the distribution of stress and displacement fields, are analogous parameters of fracture mechanics for the theory of thin inhomogeneities [55–57].

This work aims to develop an analytical and numerical method for studying the stress-strain state and strength of composites with thin deformable multilayer ribbon-like elements that are also suitable for mathematical modeling of thin inclusions with an almost arbitrary continuous thickness variation of mechanical characteristics.

#### **2. Formulation of the Problem**

We consider a structure which, following the concept of deformable solid mechanics, we will further consider as a combination of two half-spaces with elastic constants *Ek*, *νk*, *Gk* (*k* = 1, 2), at the interface of which (plane *xOz*) there is a tunnel section *L*- = [−*a*; *a*] in the direction of the shear axis *Oz* (Figure 1), in which a certain object of general thickness 2*h*(*h a*) is inserted–a package of *M* different thin plane-parallel layers - *x* ∈ *L*- ; *<sup>y</sup>* <sup>∈</sup> [*yK* <sup>−</sup> *hK*; *yK* <sup>+</sup> *hK*], *<sup>K</sup>* <sup>=</sup> 1, *<sup>M</sup>* of thickness 2*hK* 2*h* = 2 ∑*<sup>M</sup> <sup>K</sup>*=<sup>1</sup> *hK* , *y*<sup>1</sup> − *h*<sup>1</sup> = −*h*, *yM* + *hM* = *h* with orthotropic mechanical properties *GinK <sup>y</sup>* , *GinK <sup>x</sup>* in the direction of two axes (Figure 2).

**Figure 1.** Geometry and load pattern of the problem.

**Figure 2.** Multilayered inclusion.

The structure is loaded quasi-statically by shear factors (uniform shear at infinity *τ*, *τk*, concentrated forces *Qk*, and screw dislocations *bk* at points *ςk*∗), which cause longitudinal shear in the body. To ensure the straightness of the material interface at infinity, the stresses must satisfy the conditions *σ*<sup>∞</sup> *xz*2*G*<sup>1</sup> = *<sup>σ</sup>*<sup>∞</sup> *xz*1*G*2, *v*2*σ*<sup>∞</sup> *yy*−(1−*v*2)*σ*<sup>∞</sup> *xx*2 *<sup>G</sup>*<sup>2</sup> <sup>=</sup> *<sup>v</sup>*1*σ*<sup>∞</sup> *yy*−(1−*v*1)*σ*<sup>∞</sup> *xx*1 *<sup>G</sup>*<sup>1</sup> .

Let us restrict ourselves to the problem of longitudinal displacement in the direction of the *z*-axis (antiplane deformation). Then, considering that the stress-strain state (SSS) of the structure in each section perpendicular to the *z*-axis is identical, we will further consider only the plane *xOy*, which consists of two planar sections of half-spaces *Sk*(*k* = 1, 2) with a separation boundary between them in the form of abscissa axes *Ox*.

#### **3. Materials and Methods**

The construction of a mathematical model of such a layered thin inclusion-layer (internal problem) should eventually reveal the relation between the stress-strain parameters inside the inclusion and on its external surface as the influence functions *σin yz*(*x*, ±*h*), *<sup>w</sup>in*(*x*, ±*h*), which will be used in the further solution of the problem [51–53].

Let us introduce into consideration the jumps of the stress tensor components and the displacement vector for the matrix components and individual layers on *L*- :

$$\begin{aligned} \left[\sigma\_{yz}\right]\_{0,h} & \cong \sigma\_{yz1}(\mathbf{x}, -h) - \sigma\_{yz}(\mathbf{x}, h) = f\mathfrak{z}(\mathbf{x}),\\ \left[\frac{\partial \mathfrak{w}}{\partial \mathbf{x}}\right]\_{0,h} & \cong \frac{\partial \mathfrak{w}}{\partial \mathbf{x}}(\mathbf{x}, -h) - \frac{\partial \mathfrak{w}}{\partial \mathbf{x}}(\mathbf{x}, h) = \left[\frac{\sigma\_{\mathbf{x}\mathbf{z}}}{G}\right]\_{0,h} = f\_{6}(\mathbf{x}), \ \mathbf{x} \in L' \end{aligned} \tag{1}$$

$$\begin{array}{c} \left[\sigma\_{yz}^{inK}\right]\_{y\chi\_{K}h\_{K}} \cong \sigma\_{yz}^{inK}(\mathbf{x}, y\_{K} - h\_{K}) - \sigma\_{yz}^{inK}(\mathbf{x}, y\_{K} + h\_{K}) = f\_{3K}(\mathbf{x}), \; \left(\mathbf{K} = \overline{1, M}\right) \\\left[\frac{\partial \mathbf{w}^{inK}}{\partial \mathbf{x}}\right]\_{y\_{K}h\_{K}} \cong \frac{\partial \mathbf{w}^{inK}}{\partial \mathbf{x}}(\mathbf{x}, y\_{K} - h\_{K}) - \frac{\partial \mathbf{w}^{inK}}{\partial \mathbf{x}}(\mathbf{x}, y\_{K} + h\_{K}) = f\_{6K}(\mathbf{x}), \; \mathbf{x} \in L'; \end{array} \tag{2}$$

*f*3(*x*) = *f*6(*x*) = 0, *f*3*K*(*x*) = *f*6*K*(*x*) = 0, if *x* ∈/ *L*- . Hereinafter marked:

$$\left[\bullet\right]\_{y,h} = \bullet(\leftx, y - h) - \bullet(\leftx, y + h), \right]$$

$$\left\langle \bullet \right\rangle\_{y,h} = \bullet(\leftx, y - h) + \bullet(\leftx, y + h).$$

Let us analyze the methodology for constructing a mathematical model for the case of a multilayer package of thin inclusion layers. The basic relation for an arbitrary orthotropic elastic material with shear moduli *GinK <sup>x</sup>* , *GinK <sup>y</sup>* of each of the layers given by the parameters *yK*, *hK K* = 1, *M* are the equilibrium conditions:

$$\frac{\partial \sigma\_{xz}^{inK}}{\partial x} + \frac{\partial \sigma\_{yz}^{inK}}{\partial y} + \rho^K F^{inK} = 0,\tag{3}$$

where *ρ<sup>K</sup>* denotes a density of the material, and *FinK*—distribution of the mass forces, and constitutive strain-stress dependence (orthotropic linear elasticity):

$$
\sigma\_{xz}^{inK} = G\_x^{inK} \frac{\partial w^{inK}}{\partial x}, \sigma\_{yz}^{inK} = G\_y^{inK} \frac{\partial w^{inK}}{\partial y}. \tag{4}
$$

By integrating Equation (3) over the *x* limits [−*a*, *x*] and averaging, respectively, over the thicknesses of each of the heterogeneity layers *y* ∈ [*yK* − *hK*, *yK* + *hK*], we obtain:

$$\frac{1}{2h\_K} \int\_{y\_K - h\_K}^{y\_K + h\_K} \sigma\_{xx}^{inK}(\xi, y) dy \simeq \frac{1}{2} \left\langle \sigma\_{xx}^{inK} \right\rangle\_{y\_K, h\_K} = \frac{G\_x^{inK}}{2} \left\langle \frac{\partial w^{inK}}{\partial x} \right\rangle\_{y\_K, h\_K}^{} \tag{5}$$

and, accordingly, the first group of *M* equations of mathematical models of layers:

$$\frac{G\_{\text{x}}^{inK}}{2} \left< \frac{\partial w^{inK}}{\partial x} \right>\_{\text{y}\_{K,hK}} - \sigma\_{\text{xx}}^{inK}(-a) - \frac{1}{2h\_{\text{K}}} \int\_{-a}^{x} \left[ \sigma\_{\text{yz}}^{inK} \right]\_{\text{y}\_{K,hK}} (\xi) d\xi + \epsilon\_{\text{after}} + F\_{\text{after}}^{inK}(x, -h\_{\text{K}}, h\_{\text{K}}) = 0,\tag{6}$$

where *FinK aver*(*x*, <sup>−</sup>*hK*, *hK*) <sup>=</sup> *<sup>ρ</sup><sup>K</sup>* 2*hK yK*+*hK yK*−*hK <sup>x</sup>* <sup>−</sup>*<sup>a</sup> <sup>F</sup>inK*(*ξ*, *<sup>y</sup>*)*dξdy*, *K* = 1, *M* . Considering the thin-wall ratio of the inclusion layers:

$$\begin{split} & \frac{\partial w^{inK}}{\partial y} (\boldsymbol{\chi}, \boldsymbol{y}\_K + \boldsymbol{h}\_K) + \frac{\partial w^{inK}}{\partial y} (\boldsymbol{\chi}, \boldsymbol{y}\_K - \boldsymbol{h}\_K) \simeq \\ & \simeq \frac{w^{inK} (\boldsymbol{x}, \boldsymbol{y}\_K + \boldsymbol{h}\_K) - w^{inK} (\boldsymbol{x}, \boldsymbol{y}\_K - \boldsymbol{h}\_K)}{\boldsymbol{h}\_K} = -\frac{\left[w^{inK}\right]\_{\boldsymbol{y}\_K, \boldsymbol{h}\_K}}{\boldsymbol{h}\_K} \end{split}$$

$$\begin{aligned} \simeq \frac{w^{\frac{1}{\left(\lambda, \mathcal{Y}\_K \mp h\_K\right) - W} - \left(\lambda, \mathcal{Y}\_K \mp h\_K\right)}}{h\_K} &= -\frac{\lambda}{\frac{1}{h\_K}} \frac{\mathcal{Y}\_K h\_K}{h\_K}, \\\\ w^{\text{in}K}(\mathbf{x}, \mathcal{Y}\_K \mp h\_K) &= w^{\text{in}K}(\mathbf{x}, \mathcal{Y}\_K \pm h\_K) \mp 2h\_K \frac{\partial w^{\text{in}K}}{\partial \mathcal{Y}}(\mathbf{x}, \mathcal{Y}\_K \pm h\_K) \left(K = \overline{1, M}\right), \end{aligned}$$

and constitutive relations (4), we obtain the following form of the second group of *M* equations of the inclusion model:

$$-\frac{\left[w^{inK}\right]\_{y\_K,h\_K}}{h\_K} = \frac{\left\langle \sigma\_{yz}^{inK} \right\rangle\_{y\_K,h\_K}}{G\_y^{inK}} \left(K = \overline{1,M}\right),\tag{8}$$

(7)

which together with relations (6) fully describe the thin *M* -layered inclusion model written in the values of the stress-strain behavior of the inclusion package materials.

Instead of the displacement jump, in many cases, it is convenient to use the formula for the jump of the strain components:

$$\left[w^{inK}\right]\_{y\_K, h\_K}(\mathfrak{x}) = \left[w^{inK}\right]\_{y\_K, h\_K}(-a) + \int\_{-a}^{x} \left[\frac{\partial w^{inK}}{\partial \mathfrak{x}}\right]\_{y\_K, h\_K}(\mathfrak{x})d\mathfrak{x}.$$

At each inclusion layer, the balance conditions must be satisfied:

$$\int\_{-a}^{a} f\_{3K}(\xi) d\xi = -N\_{\rm xzK}(-a) + N\_{\rm xzK}(a) + 2hF\_{\rm userK}^{in}(a, h), \tag{9}$$

$$\int\_{-a}^{a} f\_{6K}(\xi) \, d\xi = \left[ w^{inK} \right]\_{y\_K, h\_K} (a) - \left[ w^{inK} \right]\_{y\_K, h\_K} (-a) \, , \tag{10}$$

where *NxzK*(±*a*) = <sup>2</sup>*hKσin* xz*K*(±*a*).

The partial cases of the model (6), (8) of the form *μinK <sup>y</sup>* = *μin <sup>y</sup>* , *μinK <sup>x</sup>* = *μin x K* = 1, *M* (all layers are the same) or *hK* → <sup>0</sup> *<sup>μ</sup>inK <sup>y</sup>* = *μin <sup>y</sup>* → 0, *<sup>μ</sup>inK <sup>x</sup>* = *μin <sup>x</sup>* → 0 *K* = 1, *M* (no inclusion or crack) or *μinK <sup>y</sup>* = *μin <sup>y</sup>* → 0 *GinK <sup>y</sup>* = *Gin <sup>y</sup>* → ∞ , *μinK <sup>x</sup>* = *μin <sup>x</sup>* <sup>→</sup> <sup>∞</sup> *GinK <sup>x</sup>* = *Gin <sup>x</sup>* <sup>→</sup> <sup>∞</sup> (perfectly rigid homogeneous inclusion) are satisfied and coincide with those known in the literature.

The solution for the matrix as an isotropic bimaterial (external problem) is obtained by the method of the problem of conjugation of analytic functions [51–53]:

$$\begin{aligned} \sigma\_{sz}(\mathbf{x}, \mathbf{y}) &= \sigma\_{sz}^{0}(\mathbf{x}, \mathbf{y}) + \mathfrak{d}\_{sz}(\mathbf{x}, \mathbf{y}), \mathbf{s} = \{\mathbf{x}, \mathbf{y}\}, \\ w(\mathbf{x}, \mathbf{y}) &= w^{0}(\mathbf{x}, \mathbf{y}) + \mathfrak{d}(\mathbf{x}, \mathbf{y}), \end{aligned} \tag{11}$$

$$\begin{array}{l} \sigma\_{yzk}(z) + i\sigma\_{xzk}(z) = \sigma\_{yzk}^{0}(z) + i\sigma\_{xxk}^{0}(z) + ip\_{k}\mathfrak{g}\_{3}(z) - \mathbb{C}g\_{6}(z) \\ (z \in S\_{k}; r = 3, 6; k = 1, 2), \\ \sigma\_{yzk}^{\pm}(x) = \mp p\_{k}f\_{3}(x) - \mathbb{C}g\_{6}(x) + \sigma\_{yz}^{0, \pm}(x), \\ \sigma\_{xzk}^{\pm}(x) = \mp \mathbb{C}f\_{6}(x) + p\_{k}g\_{3}(x) + \sigma\_{xz}^{0, \pm}(x), \\ \frac{\partial w^{\pm}}{\partial y}(x) = \mp p\_{3}(x) - p\_{3-k}g\_{6}(x) + \frac{\sigma\_{yz}^{0, \pm}(x)}{\mathbb{C}\_{k}}, \\ \frac{\partial w^{\pm}}{\partial x}(x) = \mp p\_{3-k}f\_{3}(x) + p g\_{6}(x) + \frac{\sigma\_{xz}^{0, \pm}(x)}{\mathbb{C}\_{k}}, \end{array} \tag{12}$$

where:

$$g\_r(z) \equiv \frac{1}{\pi} \int\_{L'} \frac{f\_r(\mathbf{x})d\mathbf{x}}{\mathbf{x} - z} \text{ , } s\_r(\mathbf{x}) \equiv \int\_{-a}^{\mathbf{x}} f\_r(\mathbf{x})d\mathbf{x} \text{ , } \mathbf{C} = \mathbf{G}\_{\mathbf{3} - \mathbf{k}}\\p\_k \cdot p\_k = p\mathbf{G}\_k, p = \frac{1}{\mathbf{G}\_1 + \mathbf{G}\_2}.$$

Here the upper indexes "+" and "−" correspond to the limit values of the functions at the upper and lower margins of the line *L*; the values marked with the index "0" on the top characterize the corresponding values in a solid body without modeled heterogeneities under the corresponding external load, and the values marked with the symbol "ˆ" on the top, are the perturbations of the basic stress-strain field by the presence of an inclusion [53].

The following entries [47,49] shall continue to apply:

$$\begin{array}{l} \sigma\_{yz}^{0}(z) + i\sigma\_{xz}^{0}(z) = \tau + i\{\pi\_{k} + D\_{k}(z) + (p\_{k} - p\_{j})\overline{D}\_{k}(z) + 2p\_{k}D\_{j}(z)\}, \\ D\_{k}(z) = -\frac{Q\_{k} + i\overline{\varsigma}\_{k}b\_{k}}{2\pi(z - \varsigma\_{k\*})} \left(z \in \mathbb{S}\_{k\prime} \ k = 1, 2; \ j = 3 - k\right). \end{array}$$

To connect the external and internal problems, one needs to use contact conditions between the components of the package. There are several variants of contact conditions between the layers and between the package and the matrix:

(1) Ideal (perfect) contact between all constituents of the package:

$$\begin{cases} \begin{aligned} w^{jn(K-1)}(\mathbf{x}, y\_{K-1} + h\_{K-1}) &= w^{jnK}(\mathbf{x}, y\_K - h\_K) \left(\mathbf{K} = \overline{\mathbf{2}, M}\right), \\ \sigma\_{yz}^{jn(K-1)}(\mathbf{x}, y\_{K-1} + h\_{K-1}) &= \sigma\_{yz}^{inK}(\mathbf{x}, y\_K - h\_K) \left(\mathbf{x} \in L'). \end{aligned} \end{cases} \tag{13}$$

(2) Nonperfect contact with additional tension between layers:

$$\begin{cases} \begin{aligned} w^{\dot{m}(K-1)}(\mathbf{x}, y\_{K-1} + h\_{K-1}) &= w^{\dot{m}K}(\mathbf{x}, y\_K - h\_K) \left(\mathbf{K} = \overline{\mathbf{2}, M}), \\ \sigma\_{yz}^{\dot{m}K}(\mathbf{x}, y\_K - h\_K) &= \sigma\_{yz}^{\dot{m}(K-1)}(\mathbf{x}, y\_{K-1} + h\_{K-1}) - T\_{\mathbf{K}}. \end{aligned} \end{cases} \tag{14}$$

*TK* are the surface stresses. When *TK* = 0 we have the same ideal contact (13).

(3) Contact with friction between the (*K*)-th and (*K* − 1)-th layers at the boundary {*x*, *yK* ± *hK*} in some area *x* ∈ *Lf* ⊂ *L*-

$$
\sigma\_{yz}^{in(K-1)}(\mathbf{x}, y\_{K-1} + h\_{K-1}) = \sigma\_{yz}^{inK}(\mathbf{x}, y\_K - h\_K) = -\text{sgn}\begin{bmatrix} w^{in} \\ \end{bmatrix}\_{y\_K h\_K} \mathbf{r}\_{yzK}^{\text{max}}.\tag{15}
$$

*τmax yzK* are the limit value of tangential stresses, at which slippage begins. When *<sup>τ</sup>max yzK* there is a smooth contact between these layers.

(4) Ideal contact between the boundary components of the package and the matrix:

$$\begin{cases} \begin{aligned} w^{\dot{m}1}(\mathbf{x}, y\_1 - h\_1) &= w(\mathbf{x}, y\_1 - h\_1) \\ \sigma\_{yz}^{\dot{m}1}(\mathbf{x}, y\_1 - h\_1) &= \sigma\_{yz1}(\mathbf{x}, y\_1 - h\_1), \\ w^{\dot{m}M}(\mathbf{x}, y\_M + h\_M) &= w(\mathbf{x}, y\_M + h\_M), \\ \sigma\_{yz}^{\dot{m}K}(\mathbf{x}, y\_M + h\_M) &= \sigma\_{yz2}(\mathbf{x}, y\_M + h\_M)(\mathbf{x} \in L'). \end{aligned} \end{cases} \tag{16}$$

(5) Nonperfect contact between the edge components of the package and the matrix:

$$\begin{cases} \begin{aligned} w^{in1}(\mathbf{x}, y\_1 - h\_1) &= w(\mathbf{x}, y\_1 - h\_1) \\ \sigma\_{yz}^{in1}(\mathbf{x}, y\_1 - h\_1) &= \sigma\_{yz1}(\mathbf{x}, y\_1 - h\_1) - T\_1 \\ w^{inM}(\mathbf{x}, y\_M + h\_M) &= w(\mathbf{x}, y\_M + h\_M) \\ \sigma\_{yz}^{inM}(\mathbf{x}, y\_M + h\_M) &= \sigma\_{yz2}(\mathbf{x}, y\_M + h\_M) + T\_{M+1}(\mathbf{x} \in L'). \end{aligned} \end{cases} \tag{17}$$

In the case *T*<sup>1</sup> = *TM*+<sup>1</sup> = 0 we have an ideal contact (16).

(6) Contact with friction between the inclusion and the matrix within {*x*, *y*<sup>1</sup> − *h*1}, {*x*, *yM* + *hM*} in some area *x* ∈ *Lf* ⊂ *L*-

$$\begin{cases} \sigma\_{yz1}(\mathbf{x}, y\_1 - h\_1) = \sigma\_{yz}^{in1}(\mathbf{x}, y\_1 - h\_1) = -\operatorname{sgn}(w^{in})\_{y\_1 h\_1} \tau\_{yz1}^{\max} \\ \sigma\_{yz2}(\mathbf{x}, y\_M + h\_M) = \sigma\_{yz}^{inM}(\mathbf{x}, y\_M + h\_M) = -\operatorname{sgn}(w)\_{y\_M h\_M} \tau\_{yzM}^{\max} \end{cases} \tag{18}$$

In the case *τmax yzK* there is a smooth contact.

One of the conditions (13)–(15) and one of the conditions (16)–(18) must be fulfilled simultaneously.

Using (2) and, for example, (14), we can obtain the expressions for the stresses and strains in the layers through the stress and strain limits for the inclusion package for the presence of interlayer tension:

$$\begin{split} \sigma\_{yz}^{\text{inK}}(\mathbf{x}, y\_K + h\_K) &= \sigma\_{yz}^{\text{in1}}(\mathbf{x}, -h) - \sum\_{j=1}^{K} f\_{3,j} - \sum\_{j=2}^{K} T\_j = \sigma\_{yz}^{\text{inM}}(\mathbf{x}, h) + \sum\_{j=K+1}^{M} f\_{3,j} + \sum\_{j=K+1}^{M} T\_j, \\ \frac{\partial \sigma^{\text{inK}}}{\partial \mathbf{x}}(\mathbf{x}, y\_K + h\_K) &= \frac{\partial \mathbf{w}^{\text{in1}}}{\partial \mathbf{x}}(\mathbf{x}, -h) - \sum\_{j=1}^{K} f\_{6,j} = \frac{\partial \mathbf{w}^{\text{inM}}}{\partial \mathbf{x}}(\mathbf{x}, h) + \sum\_{j=K+1}^{M} f\_{6,j}(\mathbf{x} \in L'). \end{split} \tag{19}$$

Here, the total jumps of the boundary stresses and strains for the inclusion package have the value:

$$\begin{aligned} \sigma\_{yz}^{in1}(\mathbf{x}, -\mathbf{h}) - \sigma\_{yz}^{inM}(\mathbf{x}, \mathbf{h}) &= \sum\_{j=1}^{M} f\_{3,j} + \sum\_{j=2}^{M} T\_{j\prime} \\ \frac{\partial \mathbf{w}^{in1}}{\partial \mathbf{x}}(\mathbf{x}, -\mathbf{h}) - \frac{\partial \mathbf{w}^{inM}}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{h}) &= \sum\_{j=1}^{M} f\_{6,j}(\mathbf{x} \in L^{\prime}). \end{aligned} \tag{20}$$

The resulting limit stresses and strains for the inclusion package (19), the boundary values of the stresses and strains of the matrix (12), and the boundary conditions (13)–(18) form a complete system of singular integral equations (SSIE) for the solution of the problem. Note that the dissimodularity of the inclusion layers in no way affects the peculiarity of the solution of the SSIE of the problem, which allows us to obtain a large variety of effects from manipulating the properties of the layers.

To illustrate the method, let us investigate the longitudinal shear of a structure in the form of a body with a thin two-layer inclusion with layers of thickness 2*hK*(*K* = 1, 2), 2*h* = 2*h*<sup>1</sup> + 2*h*<sup>2</sup> and orthotropic mechanical properties *GinK <sup>y</sup>* , *GinK <sup>x</sup>* , respectively, under the condition of nonperfect mechanical contact with surface tension on the contact surfaces of the structural components under different kinds of loading (Figure 3) when *<sup>ς</sup>k*<sup>∗</sup> = *xk*<sup>∗</sup> + *iyk*∗(*<sup>k</sup>* = 1, 2).

**Figure 3.** Geometry and load pattern of the problem for two-layer different-modularity thin inclusion.

According to (6), (8), the mathematical model for the two-layer inclusion is as follows:

$$\begin{cases} \begin{aligned} \mu\_{x}^{in1} \left< \frac{\partial w^{in1}}{\partial x} \right>\_{y\_{1},h\_{1}}(\mathbf{x}) - \int\_{-a}^{x} \left[ \sigma\_{xz}^{in1} \right]\_{y\_{1},h\_{1}}(\boldsymbol{\xi}) d\boldsymbol{\xi} = 2h\_{1} \left( \sigma\_{xz}^{in1}(-a) - F\_{\mbox{norm}}^{in1}(\mathbf{x}) \right), \\ \mu\_{x}^{in2} \left< \frac{\partial w^{in2}}{\partial x} \right>\_{y\_{2},h\_{2}}(\mathbf{x}) - \int\_{-a}^{x} \left[ \sigma\_{xz}^{in2} \right]\_{y\_{2},h\_{2}}(\boldsymbol{\xi}) d\boldsymbol{\xi} = 2h\_{2} \left( \sigma\_{xz}^{in2}(-a) - F\_{\mbox{norm}}^{in2}(\mathbf{x}) \right), \\ \mu\_{y}^{in1} \left< \sigma\_{yz}^{in1} \right>\_{y\_{1},h\_{1}} + \int\_{-a}^{x} \left[ \frac{\partial w^{in1}}{\partial x} \right]\_{y\_{1},h\_{1}} d\boldsymbol{\xi} + \left[ w^{in1} \right]\_{y\_{1},h\_{1}}(-a) = 0, \\ \mu\_{y}^{in2} \left< \sigma\_{yz}^{in2} \right>\_{y\_{2},h\_{2}} + \int\_{-a}^{x} \left[ \frac{\partial w^{in2}}{\partial x} \right]\_{y\_{2},h\_{2}} d\boldsymbol{\xi} + \left[ w^{in2} \right]\_{y\_{2},h\_{2}}(-a) = 0. \end{aligned} \tag{21}$$

The boundary conditions between the surfaces of the layers and the tunnel section on *L*-= [−*a*; *a*] take nonperfect with surface tension (14)–(17):

$$\begin{array}{l} \sigma\_{yz}^{in1}(\mathbf{x}, y\_1 - h\_1) = \sigma\_{yz1}(\mathbf{x}, -h) - T\_{1\prime} \\ \sigma\_{yz}^{in2}(\mathbf{x}, y\_2 - h\_2) = \sigma\_{yz1}^{in1}(\mathbf{x}, y\_1 + h\_1) - T\_{2\prime} \\ \sigma\_{yz2}(\mathbf{x}, h) = \sigma\_{yz}^{in2}(\mathbf{x}, y\_2 + h\_2) - T\_{3\prime} \\ w(\mathbf{x}, -h) = w^{in1}(\mathbf{x}, y\_1 - h\_1), \\ w^{in2}(\mathbf{x}, y\_2 - h\_2) = w^{in1}(\mathbf{x}, y\_1 + h\_1), \\ w^{in2}(\mathbf{x}, y\_2 + h\_2) = w(\mathbf{x}, h). \end{array} \tag{22}$$

Given the boundary conditions (22), the relations between jumps (1) and (2) take the form:

$$\begin{cases} f\_3(\mathbf{x}) = f\_{3,1}(\mathbf{x}) + f\_{3,2}(\mathbf{x}) + T\_1 + T\_2 + T\_{3,4} \\ f\_6(\mathbf{x}) = f\_{6,1}(\mathbf{x}) + f\_{6,2}(\mathbf{x}). \end{cases} \tag{23}$$

In addition, the integral representations of the external problem (11), (12), if the tension *TK* is not a function of the *Ox* coordinate, can be written as:

*σyz*2(*x*, *h*) = −*p*<sup>2</sup> *f*3,1(*x*) − *p*<sup>2</sup> *f*3,2(*x*) − *Cg*6,1(*x*) − *Cg*6,2(*x*)− <sup>−</sup>*p*2(*T*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*3) <sup>+</sup> *<sup>σ</sup>*0<sup>+</sup> *yz* (*x*), *σyz*1(*x*, −*h*) = *p*<sup>1</sup> *f*3,1(*x*) + *p*<sup>1</sup> *f*3,2(*x*) − *Cg*6,1(*x*) − *Cg*6,2(*x*)+ +*p*1(*T*<sup>1</sup> + *T*<sup>2</sup> + *T*3) + *σ*0<sup>−</sup> *yz* (*x*), *∂w <sup>∂</sup><sup>x</sup>* (*x*, *h*) = −*p*<sup>1</sup> *f*6,1(*x*) − *p*<sup>1</sup> *f*6,2(*x*) + *pg*6,1(*x*) + *pg*6,2(*x*)+ + *<sup>p</sup> <sup>π</sup>* (*T*<sup>1</sup> + *T*<sup>2</sup> + *T*3)*ln* <sup>1</sup>−*<sup>x</sup>* <sup>1</sup>+*<sup>x</sup>* <sup>+</sup> *<sup>σ</sup>*0<sup>+</sup> *xz* (*x*) *<sup>G</sup>*<sup>2</sup> , *∂w <sup>∂</sup><sup>x</sup>* (*x*, −*h*) = *p*<sup>2</sup> *f*6,1(*x*) + *p*<sup>2</sup> *f*6,2(*x*) + *pg*6,1(*x*) + *pg*6,2(*x*)+ + *<sup>p</sup> <sup>π</sup>* (*T*<sup>1</sup> + *T*<sup>2</sup> + *T*3)*ln* <sup>1</sup>−*<sup>x</sup>* <sup>1</sup>+*<sup>x</sup>* <sup>+</sup> *<sup>σ</sup>*0<sup>−</sup> *xz* (*x*) *<sup>G</sup>*<sup>1</sup> , (24)

which agrees well with (23).

Substituting (22)–(24) into (21) considering the expressions:

$$\begin{array}{ll} & \sigma\_{yz1}^{in1}(x, y\_1 - h\_1) = \sigma\_{yz1}(x, -h) - T\_1 \\ & \sigma\_{yz2}^{in2}(x, y\_2 + h\_2) = \sigma\_{yz2}(x, h) + T\_3 \\ & \sigma\_{yz1}^{in1}(x, y\_1 + h\_1) = \sigma\_{yz}^{in2}(x, y\_2 - h\_2) + T\_2 = \\ & = \sigma\_{yz2}^{in2}(x, y\_2 + h\_2) + f\_{3,2}(x) + T\_2 = \\ & = \sigma\_{yz2}(x, h) + f\_{3,2}(x) + T\_2 + T\_3 \\ & = \sigma\_{yz2}^{in2}(x, y\_2 - h\_2) = \sigma\_{yz1}^{in1}(x, y\_1 + h\_1) - T\_2 = \\ & = \sigma\_{yz1}^{in1}(x, y\_1 - h\_1) - f\_{3,1}(x) - T\_2 = \\ & = \sigma\_{yz1}(x, -h) - f\_{3,1}(x) - T\_2 - T\_1 \end{array} \tag{25}$$

generates the following kind of two-layer multi-module thin inclusion model in terms of jumps:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ (*p*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*1)*f*6,1(*x*) <sup>+</sup> <sup>2</sup>*p*<sup>2</sup> *<sup>f</sup>*6,1(*x*) <sup>+</sup> <sup>2</sup>*pg*3,1(*x*) <sup>+</sup> <sup>2</sup>*pg*3,2(*x*) <sup>−</sup> <sup>1</sup> *μin*<sup>1</sup> *<sup>x</sup> <sup>x</sup>* <sup>−</sup>*<sup>a</sup> <sup>f</sup>*3,1(*ξ*)*d<sup>ξ</sup>* <sup>=</sup> = <sup>2</sup>*h*<sup>1</sup> *μin*<sup>1</sup> *<sup>x</sup> <sup>σ</sup>in*<sup>1</sup> *xz* (−*a*) − *<sup>F</sup>in*<sup>1</sup> *aver*(*x*) − *<sup>∂</sup>w*<sup>0</sup> *∂x h* (*x*) <sup>−</sup> {*T*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*3} <sup>2</sup>*<sup>p</sup> <sup>π</sup> ln* <sup>1</sup>−*<sup>x</sup>* <sup>1</sup>+*<sup>x</sup>* , <sup>−</sup>2*p*<sup>1</sup> *<sup>f</sup>*6,1(*x*) <sup>+</sup> (*p*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*1)*f*6,2(*x*) <sup>+</sup> <sup>2</sup>*pg*3,1(*x*) <sup>+</sup> <sup>2</sup>*pg*3,2(*x*) <sup>−</sup> <sup>1</sup> *μin*<sup>2</sup> *<sup>x</sup> <sup>x</sup>* <sup>−</sup>*<sup>a</sup> <sup>f</sup>*3,2(*ξ*)*d<sup>ξ</sup>* <sup>=</sup> = <sup>2</sup>*h*<sup>2</sup> *μin*<sup>2</sup> *<sup>x</sup> <sup>σ</sup>in*<sup>2</sup> *xz* (−*a*) − *<sup>F</sup>in*<sup>2</sup> *aver*(*x*) − *<sup>∂</sup>w*<sup>0</sup> *∂x h* (*x*) <sup>−</sup> {*T*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*<sup>2</sup> <sup>+</sup> *<sup>T</sup>*3} <sup>2</sup>*<sup>p</sup> <sup>π</sup> ln* <sup>1</sup>−*<sup>x</sup>* <sup>1</sup>+*<sup>x</sup>* , <sup>−</sup>(*p*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*1)*f*3,1(*x*) <sup>+</sup> <sup>2</sup>*p*<sup>1</sup> *<sup>f</sup>*3,2(*x*) <sup>−</sup> <sup>2</sup>*Cg*6,1(*x*) <sup>−</sup> <sup>2</sup>*Cg*6,2(*x*) <sup>+</sup> <sup>1</sup> *μin*<sup>1</sup> *<sup>y</sup> <sup>x</sup>* <sup>−</sup>*<sup>a</sup> <sup>f</sup>*6,1(*ξ*)*d<sup>ξ</sup>* <sup>=</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> *μin*<sup>1</sup> *<sup>y</sup> win*<sup>1</sup> *y*1,*h*<sup>1</sup> (−*a*) + *T*<sup>1</sup> − *T*<sup>2</sup> − *T*<sup>3</sup> − *σ*0 *yzk <sup>h</sup>*(*x*), <sup>−</sup>2*p*<sup>2</sup> *<sup>f</sup>*3,1(*x*) <sup>−</sup> (*p*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*1)*f*3,2(*x*) <sup>−</sup> <sup>2</sup>*Cg*6,1(*x*) <sup>−</sup> <sup>2</sup>*Cg*6,2(*x*) <sup>+</sup> <sup>1</sup> *μin*<sup>2</sup> *<sup>y</sup> <sup>x</sup>* <sup>−</sup>*<sup>a</sup> <sup>f</sup>*6,2(*ξ*)*d<sup>ξ</sup>* <sup>=</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup> *μin*<sup>2</sup> *<sup>y</sup> win*<sup>2</sup> *y*2,*h*<sup>2</sup> (−*a*) + *T*<sup>1</sup> + *T*<sup>2</sup> − *T*<sup>3</sup> − *σ*0 *yzk <sup>h</sup>*(*x*). (26)

The resulting SSIE is supplemented by additional balance conditions:

$$\begin{array}{l} \int\_{-a}^{a} f\_{3,1}(\xi) d\overline{\xi} = N\_{xx}^{\text{in1}}(a) - N\_{xx}^{\text{in1}}(-a) + a(T\_1 + T\_2), \\\int\_{-a}^{a} f\_{3,2}(\xi) d\overline{\xi} = N\_{xx}^{\text{in2}}(a) - N\_{xx}^{\text{in2}}(-a) + a(T\_2 + T\_3), \end{array} \tag{27}$$

$$\begin{array}{l} \int\_{-a}^{a} f\_{6,1}(\xi) d\overline{\xi} = \left[ w^{in1} \right]\_{y\_1, h\_1} (a) - \left[ w^{in1} \right]\_{y\_1, h\_1} (-a), \\ \int\_{-a}^{a} f\_{6,2}(\xi) d\overline{\xi} = \left[ w^{in2} \right]\_{y\_2, h\_2} (a) - \left[ w^{in2} \right]\_{y\_2, h\_2} (-a), \end{array} \tag{28}$$

where *NinK xz* (*x*) = 2*hK σinK xz* (−*a*) − *<sup>F</sup>inK aver*(*x*) (*K* = 1, 2). or:

$$\begin{array}{l} \int\_{-a}^{a} f\_{3}(\xi) d\xi = N\_{xx}^{in}(a) - N\_{xx}^{in}(-a) + a(T\_{1} + T\_{3}),\\ \int\_{-a}^{a} f\_{3,1}(\xi) d\xi + \int\_{-a}^{a} f\_{3,2}(\xi) d\xi + 2a(T\_{1} + T\_{2} + T\_{3}) = \\ = N\_{xx}^{in1}(a) - N\_{xx}^{in1}(-a) + a(T\_{1} + T\_{2}) + N\_{xx}^{in2}(a) - N\_{xx}^{in2}(-a) + a(T\_{2} + T\_{3}),\end{array} \tag{29}$$

$$\int\_{-a}^{a} f\_{3,1}(\xi) d\xi + \int\_{-a}^{a} f\_{3,2}(\xi) d\xi = N\_{xx}^{\text{in1}}(a) - N\_{xx}^{\text{in1}}(-a) + N\_{xx}^{\text{in2}}(a) - N\_{xx}^{\text{in2}}(-a) - a(T\_1 + T\_3).$$

To preserve the quasi-static equilibrium of the considered microstructure, one should also require the fulfillment of the condition of the balance of surface forces *T*<sup>1</sup> + *T*<sup>3</sup> = 2*T*2.

The resulting system of Equations (26)–(29) is reduced to a system of linear algebraic equations concerning the unknown coefficients of the decomposition of the desired influence functions [51–53] into a series by Jacobi-Chebyshev polynomials, described in [55].

#### **4. Numerical Results and Discussion**

In fracture mechanics, it is common to use the stress intensity factor (SIF) *K*<sup>3</sup> to describe the asymptotics of the SSS in the vicinity of the crack tip [8,38]. For the case of a thin elastic inclusion, this is not sufficient [56]. The introduction of a system of polar coordinates (*r*, *θ*) with the origin near the right or the left tip of the inclusion *z*<sup>1</sup> = ±*rexp*(*iθ*) ± *a* (Figure 3), makes it possible to obtain two-term asymptotic expressions for the distribution of the stresses and displacements in the vicinity of the tips (|*z*1| 2*a*) [55] using the generalized stress intensity factors (GSIF) introduced by the expression:

$$K\_{31} + iK\_{32} = \lim\_{r \to \infty (\theta = 0, \pi)} \sqrt{2\pi r} \left(\sigma\_{yz} + i\sigma\_{xz}\right).$$

Consider also the following dimensionless values, marked with a "~" at the top, which significantly reduce the number of calculations without loss in generality:

$$\begin{array}{l} \widetilde{\boldsymbol{x}} = \frac{\boldsymbol{x}}{\widetilde{\boldsymbol{h}}} \widetilde{\boldsymbol{K}} = \frac{\boldsymbol{h}\_{k}}{\widetilde{\boldsymbol{a}}\_{r}} \widetilde{\boldsymbol{y}} = \frac{\boldsymbol{y}}{\widetilde{\boldsymbol{a}}\_{r}} \widetilde{\boldsymbol{\tau}}\_{k} = \frac{\widetilde{\boldsymbol{\tau}}\_{k}}{\widetilde{\boldsymbol{G}}\_{\operatorname{gav}}}, \widetilde{\boldsymbol{\tau}} = \frac{\widetilde{\boldsymbol{\tau}}}{\widetilde{\boldsymbol{G}}\_{\operatorname{gav}}}, \\\ \widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}}^{\rm{in}K}(\widetilde{\boldsymbol{x}}) = \frac{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}}^{\rm{in}K}(\boldsymbol{x})}{\widetilde{\boldsymbol{G}}\_{\operatorname{gav}}} \widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{y}}}^{\rm{in}K}(\widetilde{\boldsymbol{x}}) = \frac{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{y}}}^{\rm{in}K}(\boldsymbol{x})}{\widetilde{\boldsymbol{G}}\_{\operatorname{gav}}}, \\\ \widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}} = \frac{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}} \widetilde{\boldsymbol{w}}\_{\boldsymbol{x}} (\boldsymbol{k} - \boldsymbol{1}\mathcal{L})}{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}} \widetilde{\boldsymbol{w}}} \widetilde{\boldsymbol{p}}\_{\widetilde{\boldsymbol{x}}} = \boldsymbol{p}\_{k}, \widetilde{\boldsymbol{C}} = \frac{\widetilde{\boldsymbol{C}}}{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}} \boldsymbol{w}}, \\\ \widetilde{\boldsymbol{w}}\_{\mathcal{X}}(\widetilde{\boldsymbol{x}}) = \frac{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}}(\boldsymbol{x})}{\widetilde{\boldsymbol{G}}\_{\widetilde{\boldsymbol{x}}} \widetilde{\boldsymbol{w}}} \widetilde$$

where *K*<sup>+</sup> <sup>31</sup>, *<sup>K</sup>*<sup>+</sup> <sup>32</sup>—GSIF's near the tip of inclusion *x* = +*a*.

The use of dimensionless values will make it possible to interpret the obtained quantitative results and qualitative conclusions on any variant of specific materials of inclusion layers or matrix by simple recalculation due to the universality of the mathematical model of a thin deformable inclusion and the method of problem-solving. The investigation of the influence of the inclusion layers different modularity, external force factors at non-ideal contact with the surface tension of the structural components on the unmeasured stressstrain field parameters on the inclusion surfaces, and the dimensionless stress intensity factor *K*<sup>31</sup> are illustrated in Figures 4–16. Figures 4–7 shows the results of a study of the stress distributions on the contact surfaces and displacement jumps on the inclusion as a function of the degree of dissimilarity of the inclusion layers under different external loads (Figures 4 and 5 illustrate the effect of a far-field uniform shear loading, and Figures 6 and 7 illustrate the effect of a concentrated force on similar structures) and in the absence of surface forces.

When one of the layers is significantly softer than the matrix, the effect of "unloading" (stress level reduction) of the surfaces is observed irrespective of the stiffness of the second layer. And this effect is more local than the loading by the concentrated forces located in the points *<sup>ς</sup>k*<sup>∗</sup> = *xk*<sup>∗</sup> + *iyk*∗; *xk*<sup>∗</sup> = 0, *<sup>y</sup>*2<sup>∗</sup> = −*y*1<sup>∗</sup> = *<sup>d</sup>* of order *<sup>d</sup>*/*<sup>a</sup>* ≈ *<sup>O</sup>*(1). Figures 4 and 5 reflect the known fact that the stress variation on most of the inclusion surfaces is small and changes abruptly as they approach the tips. In contrast, the applied near the inclusion concentrated force (Figures 6 and 7) essentially perturbs the character of stress distribution along the inclusion axis, its maximum value for such a loading is reached on the geometric symmetry axis of the problem. With the removal of the point of force application (increase

in *d*) the character of the stresses changes approaches the characteristic of a far-field uniform shear loading (Figures 4 and 5). Figure 7d illustrates the proportionality of the displacement jumps of each layer to their stiffness.

**Figure 4.** Stress distribution along with the upper interface (layer 2 of the inclusion—matrix halfspace *S*2) (**a**); the boundary between layers (layer 1–layer 2) (**b**); lower interface (layer 1—matrix half-space *S*<sup>1</sup> ) (**c**), and the displacement jump on the inclusion (**d**) for a layer 1 stiffer than the matrix as a function of the change in stiffness of layer 2 under the load uniformly distributed at infinity.

**Figure 5.** Stress distribution along with the upper interface (layer 2 of the inclusion—matrix halfspace *S*2) (**a**); the boundary between layers (layer 1–layer 2) (**b**); lower interface (layer 1—matrix half-space *S*<sup>1</sup> ) (**c**), and the displacement jump on the inclusion (**d**) for a layer 1 softer than the matrix as a function of the change in stiffness of layer 2 under the load uniformly distributed at infinity (1—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 6.** Stress distribution along with the upper interface (layer 2 of the inclusion–matrix halfspace *S*2) (**a**); the boundary between layers (layer 1–layer 2) (**b**); lower interface (layer 1—matrix half-space *S*<sup>1</sup> ) (**c**), and the displacement jump on the inclusion (**d**) for a layer 1 stiffer than the matrix as a function of the change in stiffness of layer 2 under the load by concentrated forces at points *<sup>ς</sup>k*<sup>∗</sup> = *xk*<sup>∗</sup> + *iyk*∗; *xk*<sup>∗</sup> = 0, *<sup>y</sup>*2<sup>∗</sup> = −*y*1<sup>∗</sup> = *<sup>d</sup>*.

**Figure 7.** Stress distribution along with the upper interface (layer 2 of the inclusion–matrix halfspace *S*2) (**a**); the boundary between layers (layer 1–layer 2) (**b**); lower interface (layer 1—matrix half-space *S*<sup>1</sup> ) (**c**), and the displacement jump on the inclusion (**d**) for a layer 1 softer than the matrix as a function of the change in stiffness of layer 2 under the load by concentrated forces at points *<sup>ς</sup>k*<sup>∗</sup> = *xk*<sup>∗</sup> + *iyk*∗; *xk*<sup>∗</sup> = 0, *<sup>y</sup>*2<sup>∗</sup> = −*y*1<sup>∗</sup> = *<sup>d</sup>* (1—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 8.** Influence of the level of dissimodularity on the GSIF *K*<sup>31</sup> under the load by uniformly distributed on infinity stress and absence of surface tension.

**Figure 9.** Influence of the level of dissimodularity on the GSIF *K*<sup>31</sup> under the load by uniformly distributed on infinity stress and absence of surface tension.

**Figure 10.** The effect of changing the distance *d* of the point of application of concentrated forces from the inclusion and the level of dissimodularity to the GSIF *K*<sup>31</sup> in the absence of surface tension and layer 1 softer from the matrix (4—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 11.** The effect of changing the distance *d* of the point of application of concentrated force from the inclusion and the level of dissimodularity to the GSIF *K*<sup>31</sup> in the absence of surface tension and equivalent to the matrix material layer 1 (2—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 12.** Influence of the level of dissimodularity of the inclusion layer materials on the GSIF *K*<sup>31</sup> in the absence of surface tension (3—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 13.** The effect of changing the level of dissimodularity of the inclusion layer materials on the GSIF *K*<sup>31</sup> in the absence of surface tension under a concentrated force loading.

**Figure 14.** Influence of interlayer surface tension on the GSIF *K*<sup>31</sup> for the same inclusion layer materials when loaded by a uniformly distributed stress at infinity (3—result for the case of the same layer materials, verified by comparison with [50,53]).

**Figure 15.** Influence of the interlayer surface tension for the same materials of the inclusion layers when loading with a concentrated force at the point *ς*2∗.

**Figure 16.** The effect of the interlayer surface tension on the GSIF *K*<sup>31</sup> when layer 2 is softer than the matrix, and the layer 1 is of the arbitrary material under the loading by stress uniformly distributed at infinity; (6—result for the case of the same layer materials verified by comparison with [50,53])).

Figures 8–13 show the results of the study of the effect of the level of dissimodularity on the GSIF *K*<sup>31</sup> under different external loading and in the absence of surface forces. It is noteworthy that the increase in the level of dissimodularity of the materials of the inclusion layers significantly affects the GSIF when the stiffness of one of the layers is greater than that of the matrix (Figures 8, 9, 12 and 13) regardless of the type of loading.

Figures 10 and 11 confirms the known effect of the GSIF *K*<sup>31</sup> maximum under the loading by a concentrated forces placed at a distance approximately *d* ≈ *a* from the inclusion axis, irrespective of the stiffness of the materials of the layers. However, it is more pronounced in the stiffness range of the materials of the layers softer than the matrix material. Moreover, if the material of one of the layers is equivalent to that of the matrix (Figure 11), we obtain the known results for a homogeneous elastic inclusion at the interface of the matrix materials [50–53].

Figures 14–16 show the results of the study of the influence of the level of dissimodularity of the inclusion layers and the presence of surface forces under different external loads on the GSIF *K*31. It was found that the presence of surface forces leads to an increase in the SIF if they are directed toward the external load, and a decrease if they are directed in the opposite direction from the external load (Figures 14 and 15). The dissimodularity of the materials of the layers significantly distorts this effect, which is especially noticeable when one of the layers is significantly softer than the matrix material. It is revealed that there are certain combinations of external load parameters, surface forces, and material properties of the layers, at which there are local SIF extremes. This effect can be useful in designing the modes of operation of structures with such a structure.

The solution method and results obtained for the two-layer inclusion have been verified by the coincidence of the numerical results with those known in literature [47,50–53] for a homogeneous thin elastic inclusion—the curves 1 in Figures 5 and 7; 3 in Figure 8; 4 in Figure 10; 2 in Figure 11; 3 in Figure 12; 3 in Figures 14 and 15; 6 in Figure 16.

#### **5. Conclusions**

A mathematical model of a thin multilayer inclusion of finite length with orthotropic properties of the layers is constructed, taking into account the effect of surface energy on their interfaces. On its basis, we derive a system of equations for solving the problems of antiplane deformation of a bimaterial with thin multilayer interfacial linearly elastic inclusions under arbitrary force and dislocation loading in the case where the inclusionmatrix contact may be ideal or with surface tension or sliding (smooth or frictional).

It is found that for the corresponding problems with bilayer inclusions:


All these conclusions may be useful for the design of a layered inclusion and the modes of operation of such structures. The proposed method has proved to be effective for solving a whole class of strain problems for bodies with thin deformed inclusions of finite length and may be used for the calculation of FGM inclusions. The addition of the proposed method by the homogenization method [58] will give an obvious opportunity to solve the problem of thin deformable heterogeneous inclusions in periodically layered composites.

**Author Contributions:** Conceptualization, Y.P., H.S., V.H.; methodology, Y.P., H.S.; software, Y.P., L.P.; validation, Y.P., H.S.; formal analysis, Y.P.; investigation, Y.P., H.S; resources, V.H.; data curation, Y.P.; writing—original draft preparation, Y.P.; writing—review and editing, Y.P.; visualization, Y.P, L.P.; supervision, H.S.; project administration, V.H.; funding acquisition, V.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are openly available at https://doi. org/10.3390/ma14174928, reference number [51], https://doi.10.2478/ama-2018-0029, reference number [52].

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


#### **References**


**Xinna Liu 1, Shuai Zhang 2,\*, Yanmei Bao 2, Zhongran Zhang 3,\* and Zhenming Yue <sup>2</sup>**

<sup>1</sup> Rongcheng Campus, Harbin University of Science and Technology, Rongcheng 264300, China


**\*** Correspondence: 202037591@mail.sdu.edu.cn (S.Z.); zzhongran@163.com (Z.Z.)

**Abstract:** Based on the twin bridge shear specimen, the cyclic shear experiments were performed on 1.2 mm thin plates of 316L metastable austenitic stainless steel with different strain amplitudes from 1 to 5% at ambient temperature. The fatigue behavior of 316L stainless steel under the cyclic shear path was studied, and the microscopic evolution of the material was analyzed. The results show that the cyclic stress response of 316L stainless steel exhibited cyclic hardening, saturation and cyclic softening, and the fatigue life is negatively correlated with the strain amplitude. The microstructure was analyzed by using electron back-scattered diffraction (EBSD). It was found that grain refinement and martensitic transformation during the deformation process led to rapid crack expansion and reduced the fatigue life of 316L.

**Keywords:** cyclic shear; strain amplitude; cyclic response; martensitic transformation

#### **1. Introduction**

With excellent strength, corrosion resistance, heat resistance, ductility and machinability, 316L stainless steel has been widely used in industrial applications [1–3]. The steel is usually made into parts that are subjected to certain cyclic loads at low, ambient or high temperatures, which will lead to fatigue failure of the parts [4]. Therefore, in order to improve the service life of materials under cyclic loading conditions, many scholars have continued to conduct in-depth studies on the microstructure and deformation mechanism of materials under cyclic loading. Pham et al. [5] conducted axial fatigue loading experiments on AISI 316L at room temperature, and found that the cyclic deformation response can be divided into three stages: cyclic hardening, followed by cyclic softening, and, finally, cyclic saturation before failure. In total, 316L stainless steel exhibits a cyclic hardening softening response in strain-controlled cyclic torsion [6]. The cyclic stress behavior of 316LN is determined mostly by the internal stress under non-proportional loading, and the initial cyclic hardening is greatly enhanced by the increase in loading non-proportionality [7]. Facheris et al. [8] carried out a set of uniaxial, torsional and multiaxial low-cycle fatigue and strain-controlled ratcheting tests. The effect of ratcheting on the mechanical response was found to be quantitatively stronger, causing a more pronounced drop of fatigue life. Zhou et al. [9] developed a cyclic constitutive model implemented to describe the cyclic behaviour of 316L including the hardening/softening and the strain range memory effect. During the uniaxial ratchetting deformation of 316L stainless steel, the dislocation density increases progressively, and twinning and strain-induced martensitic phases do not occur within the specified number of cycles [10]. Observations of the life-terminated dislocation arrangements by transmission electron microscopy showed that the dislocation microstructure depends essentially on the plastic strain amplitude, which in turn is strongly correlated to the stress amplitude and average stress [11]. It is believed that the austenite phase in some stainless steels is in a metastable state at ambient temperature due to the

**Citation:** Liu, X.; Zhang, S.; Bao, Y.; Zhang, Z.; Yue, Z. Strain-Controlled Fatigue Behavior and Microevolution of 316L Stainless Steel under Cyclic Shear Path. *Materials* **2022**, *15*, 5362. https://doi.org/10.3390/ma15155362

Academic Editors: Grzegorz Lesiuk and Dariusz Rozumek

Received: 9 July 2022 Accepted: 24 July 2022 Published: 4 August 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

amount of Cr and Ni. In the process of monotonic or cyclic deformation, the deformation of the metastable austenite leads to the phase transformation from initial fcc austenite (γ) to final stable martensite(α- ) [12–15]. Li et al. [16] performed a series of cyclic tests on 304L stainless steel with different loading paths, which showed that the growth rate of martensite content became higher and its distribution in the austenitic matrix was more uniform as the loading path increased non-proportionally. The sensitivity of the martensite induced by deformation depends on the chemical composition, temperature, degree of plastic deformation, and strain rate [17,18]. At the same time, strain-induced martensitic phase transformation can affect the ductility of austenitic stainless steels. It is believed that austenitic stainless steels exhibit significant work hardening, leading to the transformation from austenite to martensite [19]. Okayasu et al. [20] conducted an in-depth study of monotonic loading of austenitic steels at room and low temperatures and modeled the formation of martensite due to deformation. Das et al. [21] studied the effect of strain rate on strain induced martensite transformation characteristics. Some scholars also predicted the life of materials under fatigue loading. Branco et al. [22] developed the total strain energy density method to evaluate the fatigue life of notched specimens subjected to multiaxial loads, and the fatigue master curve can be efficiently generated from only two uniaxial strain-controlled tests, and a set of numerical simulations performed via single-element elastic-plastic models. Pelegatti et al. [23] established a robust procedure for durability design to estimate the strain–life curve, which was an effective way to deal with durability problems. Li et al. [24] proposed an energy-based model to predict the creep-fatigue, combined with high-low cyclic loading life. Because the fatigue fracture surface morphology can provide additional information for the analysis, many people have made a detailed analysis of the fracture surface. Fatigue fracture morphology is studied at different scales, and the most common fracture studies are observed using scanning electron microscopy (SEM) [25,26]. Post-failure applications of fracture surface measurements help to study the fatigue life and the fatigue damage accumulation in notched specimens subjected to severe stress gradient effects [27]. Macek et al. [28] studied the effect of asynchronous axial torsional strain-controlled on the fracture surface behavior of thin-walled tubular austenitic steel specimens. They also found that features of the post-failure fractures were related to the loading conditions and the fatigue life and the loading path significantly affects the surface topography. One of their conclusions is that the total volume fraction of martensite decreases when the sample temperature and the strain rate are increased. However, to date, although most researchers have studied the effects of stretching or combined torsion and tension on the microstructure and fracture behavior of austenitic stainless steel, few studies have studied the fatigue behavior of austenitic stainless steels under cyclic shear path.

Various forms of shear specimens have been used in the field of materials research, mainly applied to analyze forming properties and fracture behavior. Although the use of shear specimen's own geometry in the universal testing machine can achieve shear loading, it will produce a reaction torque in the process of loading, making the specimen deformation, resulting in shear path change [29,30]. Yin et al. [31] proposed the specimen for in-plane torsional experiments, which used the thinning method of machining grooves to form the experimental area. The shear loading is then completed by the relative rotation of the inner and outer rings. However, due to the need for mechanical machining of the specimen, there must be a hardening layer of interference, and the uniformity of the machined thickness cannot be guaranteed during the machining of the thin plate. Brosius et al. [32] proposed a new twin bridge shear specimen which can achieve large shear deformation by shearing the middle experimental region through the relative rotation of the inner and outer rings, and the stress-train change is measured by the torque sensor and the rotation angle. The experimental process does not have the high response movement of the unilateral shear specimen, no compensation is required, and the two experimental regions of the twin bridge shear specimen have the same deformation direction, which is suitable for studying the plate anisotropy.

Therefore, in this study, the principle of the twin bridge shear experiment is chosen to optimize the design of a shear fatigue test. Extensive strain-controlled experiments were conducted on 316L austenitic stainless steel by means of a designed cyclic shear fatigue testing machine. The main purpose is to investigate how deformation-induced martensite affects cyclic deformation and fatigue life, and to evaluate the effect of cyclic shear on the microstructural evolution of 316L austenitic stainless steel by microscopic analysis of test samples.

#### **2. Experimental Procedure**

#### *2.1. Experimental Setup and Specimens*

To study the fatigue behavior of materials under the cyclic shear path, a cyclic shear fatigue testing machine based on the twin bridge shear specimen was designed. The physical drawing of the machine and the geometry of the test piece is shown in Figure 1. The reducer in the base provides power to the power shaft so that the power shaft can rotate. The torque sensor is fixedly connected with the reducer housing, so that the Torque sensor is relatively static with the base. The outer and inner rings of the specimen are in contact with the upper surface of the torque sensor and the upper surface of the power shaft, respectively. The middle pressing structure passes through the circular hole of the middle clamping plate and the central hole of the test piece in turn, and is connected with the power shaft through threads. The inner ring of the specimen is firmly fixed to the power shaft by rotating the bolt on the middle pressing structure. When the power shaft rotates, the inner ring of the specimen rotates with it. The ring-pressing structure is fixedly connected with the torque sensor by threads. The outer ring of the specimen is firmly fixed to the torque sensor by rotating the bolt on the ring-pressing structure. Therefore, when the reducer rotates, the power shaft and the inner ring of the specimen rotate together, but the outer ring of the specimen remains relatively stationary with the equipment, thus making the inner and outer rings of the specimen rotate relatively, and then completing the shear loading of the specimen. The connecting shaft is fixedly connected with the middle pressing structure through threads. The power shaft drives the connecting shaft to rotate, thus ensuring that the connecting shaft will not be deformed by the torsional force during the loading process. The amplifying disc is fixedly connected with the connecting shaft and rotates together with it, which can further improve the resolution of angle detection. One end of the encoder is fixed on the ring-pressing structure by magnetic suction, and the other end is in contact with the amplifying disc, so that the encoder can be driven to rotate when the amplifying disc rotates, thus recording the rotation angle. The specimen fixation method adopted by this device will not cause large deformation of the positioning piece due to the influence of external force, nor will it cause deformation of the key structure due to the action of torque, to better ensure the consistency of the experimental results [33]. At the same time, because one end of the encoder is fixed to the ring-pressing structure, and the other end is in contact with the amplification disc, this connection method effectively shields the impact of the transmission gap of the system on the experimental accuracy, so that the angular resolution can reach 0.002◦.

#### *2.2. Experimental Materials and Methods*

The commercial 316L metastable austenitic stainless steel sheet produced by the hot rolling process in this study has excellent heat resistance, ductility, corrosion resistance, and mechanical properties. The specimen used for the fatigue test is shown in Figure 1b. Figure 2 shows the metallographic image of the material with 98% austenite content. The chemical composition of the specimen measured by using EBSD is shown in Table 1.

**Figure 1.** (**a**) Cyclic shear fatigue testing machine, (**b**) twin bridge shear specimen.

**Figure 2.** Microstructure of 316L before fatigue experiments.

**Table 1.** Chemical composition in weight percentage of the AISI 316L hot rolled plate.


In order to verify the relationship between the rotation angle of the inner and outer rings and the strain, the monotonic shear loading experiment of 316L was carried out by using DIC. The angle acquisition of DIC and the encoder of the testing machine adopt the same time interval and start at the same time. The relationship between strain and angle is determined by interpolation [34]. The geometry of the 316L tensile specimen is shown in Figure 3a. The equivalent shear stress–strain and tensile stress–strain curves of 316L is shown in Figure 3b. It can be seen from the figure that the yield point and strengthening process of 316L are very similar under the tensile and shear paths, and stretching has better elongation. At the stage of failure, the curve decreases rapidly under the tensile state and slowly under the shear state. The von Mises stress and strain studied in this paper were regarded as equivalent stress and strain, and the von Mises equation was used to calculate the equivalent stress and strain.

**Figure 3.** (**a**) Specimen geometry in monotonic tensile test, (**b**) comparison of tensile and shear, (**c**) relationship between equivalent strain and rotation angle.

From Figure 3c, a linear relationship between the rotation angle and the equivalent strain can be observed, which is consistent with the research conclusion of Yin et al. [34]. Therefore, the results of Figure 3c can be used to determine the relationship between the magnitude of strain amplitude and the loading angle in subsequent studies. At room temperature, the strain rate (dε/dt) was constant at 2 × <sup>10</sup>−<sup>2</sup> <sup>S</sup>−<sup>1</sup> and the strain-controlled symmetric shear loading tests were carried out under different total strain amplitudes (ε/2) ranging from 1 to 5%, the strain ratio was −1 (R = εmax/εmin = −1). Three tests were carried out under each strain condition, and a total of 15 groups of tests were carried out. After the fatigue test, the test section in the specimen was cut off by wire cutting. Then specimens were ground and polished with sandpaper of different specifications, and polished with 0.5 μm diamond polishing fluid and polishing cloth until the surface of the specimen had no scratches. Finally, these specimens were electropolished at room temperature. EBSD was used to analyze the original specimens and cracked specimen. The EBSD data were processed by ATEX software without data cleaning to obtain microstructure in different states.

#### **3. Experimental Results and Analysis**

Figure 4a,b show the stress response at strain amplitudes of 1% and 5%. The initial loading direction was defined as the positive direction of stress and strain. It can be seen from the figure that the distribution of the maximum positive stress and negative stress is approximately symmetrical under the same strain amplitude. The stress increases obviously at the initial stage of loading. With the increase in the number of cycles, the material was continuously hardened, then continuously softened until failure. Figure 4c shows the variation curves between angle and torque when the strain amplitude was 1%. The changes in shape are caused by the continuous hardening of the material, followed by continuous softening until failure. With the increase in the number of cycles, the strength of the material gradually increases, making the hysteretic curve gradually steeper. When the material hardens to a certain extent, the strength of the material begins to decrease gradually, and the hysteretic curve flattens gradually. The relationship between the stress amplitude and the number of cycles N under five different strain amplitudes is plotted in Figure 4d. It can be seen from the figure that the material hardens rapidly at the beginning of loading and the hardening amplitude increases with the increase in the strain amplitude, that is, the cyclic hardening of 316L is affected by the loading amplitude. With the increase in the strain amplitude, the stress amplitude increases, and the fatigue life decreases.

**Figure 4.** (**a**) Stress changes during 1% strain amplitude loading, (**b**) stress changes during 5% strain amplitude loading, (**c**) curves of relative rotation angle of specimen and torque under 1% strain amplitude, (**d**) cyclic stress response curves with different strain amplitudes.

The change rate of the cyclic stress response curve was used to analyze the stress change rate. The fatigue life of the specimen was defined as the number of cycles, and the first derivative of the stress amplitude in the fatigue life was carried out. The relationship between the stress change rate and the number of cycles under different strain amplitudes is shown in Figure 5a. As shown in Figure 5b, the fatigue life is divided into four stages by using 1% strain amplitude as an example: rapid change period, stabilization period, transition period, and failure period. Half the number of cycles at this failure point is defined as the half-life. In the rapid change period, the stress change rate decreases rapidly and tends to be stable gradually. The stabilization period includes the cyclic hardening

and softening of the material. The hardening stage is above the zero axis and the softening stage is below the zero axis as shown in Figure 5b. With the increase in the number of cycles, the stress change in the hardening stage decreases linearly and the stress change rate in the softening stage increases linearly. The proportion of the stabilization period in the whole life cycle decreases with the increase in the strain amplitude. The stabilization period accounts for 76.5% of the whole life cycle during 1% strain amplitude. In other words, the length of the stabilization period determines the length of fatigue life. The longer the proportion of stabilization period, the longer the fatigue life. During the transition period, the stress change rate increases rapidly. At this time, obvious cracks have appeared in the specimen and the crack growth rate continues to increase. Finally, the material reaches the failure point, the specimen enters the failure period, and the stress change rate decreases rapidly, which indicate the fracture of the specimen.

**Figure 5.** The change rate of the cyclic stress response curve of fatigue life: (**a**) the change rate of stress response curve under different strain amplitudes, (**b**) the change rate of cyclic stress response curves during 1.0% strain amplitude.

To further study the cyclic hardening and softening behavior of materials under different strain amplitudes, the cyclic hardening ratio (HR) and cyclic softening ratio (SR) are used to describe this phenomenon [33,35]. Figure 6 shows the relationship between the strain amplitude and cyclic hardening and softening. Under different strain amplitudes, the hardening ratio of the material increases with the increase in the strain amplitude and the softening ratio of the material almost remains unchanged. It is shown that the softening ability of 316L has little relation to the strain amplitude, and the strain amplitude affects the hardening ability of 316L.

**Figure 6.** Cyclic hardening and softening ration under different strain amplitudes.

From the previous cyclic fatigue stress change diagram, it can be seen that the distribution of positive stress and negative stress is approximately symmetrical. In order to compare the stress change under cyclic fatigue with those under monotonic shear, the positive stress under cyclic fatigue was selected for analysis. The maximum positive stress at each strain amplitude was connected, which was defined as the maximum hardening curve (MH). The positive stress at the failure point at each strain amplitude was connected, which was defined as the failure strength curve (FS). According to the number of cycles at the failure point, the number of cycles corresponding to the half-life was calculated. Connecting the stress corresponding to the half-life of different strain amplitudes defined as the half-life stress curve (HS), MS, MH, HS, and FS were plotted in Figure 7. It can be seen from the figure that MH, HS, and FS are all approximately linearly related to the strain amplitude, and MH, HS, and FS are all higher than MS, which is caused by the hardening of the material under cyclic shear loading. MH and HS basically coincide at the same strain amplitude, which indicates that the hardening degree of 316L reaches the maximum at the half-life. To prove this point, the hysteresis curve at 1% strain amplitude is used as an example, as shown in Figure 8. It can be seen that the stress–strain curve at half-life is located at the maximum stress position of the entire hysteresis curve, which is the time of maximum hardening. In other words, the hardening and softening of 316L account for half of the fatigue life, respectively. Therefore, the strain amplitude under cyclic shear fatigue has a very important effect on the hardening and fatigue life of 316L. The fatigue life is approximately linear with the strain amplitude, and decreases linearly with the increase in the strain amplitude, as shown in Figure 9.

**Figure 7.** Monotonic shear curve, maximum hardening, half-stress, and failure strength relationship for 316L under cyclic shear path.

**Figure 8.** Hysteresis curve at 1% strain amplitude.

**Figure 9.** Fatigue life at different strain amplitude.

#### **4. Microstructure Evolution during Cyclic Shear Loading**

The EBSD technique was used to analyze the metallographic composition of the test specimens under cyclic fatigue shear path, and Figure 10 shows that the original material of 316L austenitic stainless steel was mainly austenite. The specimen produced a crack under the shear path, which was surrounded by martensite and the grain was refined. This means that the strain-induced austenite transforms into martensite with the crack initiation, and results in grain refinement around the crack. The tendency to the strain-induced martensite transformation has been explained in terms of the variation in the chemical free-energy difference between the austenite and martensite phases, referred to as the chemical driving force [18]. The hardening of austenitic stainless steels caused during fatigue cycle under a push-pull path is mainly due to deformation-induced martensite and dynamic strain aging (DAS). At room temperature, the decrease in fatigue life is due to the rapid crack growth induced by deformation-induced martensite hardening. The amount of martensite increases with the increase in strain amplitude, which is responsible for the rapid secondary hardening. Because no martensite was found at high temperature, the hardening was caused by dynamic strain aging [36]. Therefore, the hardening of 316L under cyclic shear path at room temperature is caused by martensite transformation. With the increase in cycle times, the number of martensite increases, which makes the crack propagate rapidly and leads to the decrease in fatigue life.

**Figure 10.** EBSD phase image of specimen fatigued cycle. (**a**) Phase image of the original specimen, (**b**) phase image of the specimen after experiment.

#### **5. Conclusions**

The strain-controlled fatigue properties of 316L austenitic stainless steel was investigated and the cyclic deformation microstructure was analyzed by EBSD in this work. The conclusions are as follows:


In conclusion, this study provided an in-depth investigation of the failure mechanism of 316L under shear cyclic path through experiments and microstructure observation, which provided a pre-theoretical guidance for the development of a life prediction model under cyclic shear path. The failure scenarios at different strain ratios and the development of life prediction models will be carried out in future work.

**Author Contributions:** Conceptualization, X.L.; Data curation, X.L. and S.Z.; Formal analysis, X.L.; Funding acquisition, Z.Y.; Investigation, X.L. and S.Z.; Methodology, Z.Z.; Project administration, Z.Z.; Software, Z.Z. and S.Z.; Validation, Y.B. and Z.Y.; Writing—original draft, X.L. and S.Z.; Writing review and editing, Y.B. and Z.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the financial support from National Natural Science Foundation of China (NO. 52175337 and 51975327).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

