# Rigid line – Hire Academic Expert

**Rigid line inclusion in a matrix – simulation**

**Keywords:** Line inclusions, composite materials, stress concentration

**1. ****Background****/****Introduction**

A rigid line inclusion is a stiffener dispersed within a matrix material. Mathematically, it can be modeled as a narrow and infinitely rigid reinforcement. The rigid line model was applied to various classic elastic mechanics problem such as load diffusion (Koiter, 1955) and bi-material interface (Ballarini, 1990). It has been experimentally demonstrated that strong stress concentrations occur at stiff inclusion edges (Misseroni et al., 2014; Noselli et al., 2010). Therefore, many authors have modeled the singularities of the stresses and strains around the rigid inclusion tips. Kerr et al. (1998) studied the anti-plane problem of a two-dimensional strip, reinforced by a periodic array of rigid inclusions that are perpendicular to the faces of the strip. The equivalent shear modulus of the medium and the singularity of the stress at the inclusion tips were calculated for various reinforcement geometries. Tang et al. (1997) solved the interaction of a rigid line inclusion with an elastic circular inclusion. They obtained the singular stresses at the tips of the rigid line inclusion and the stresses at the interface of the elastic inclusions. The interaction of parallel, collinear and radial line inclusions have been studied with the help of the boundary element method by Jobin et al. (2019). In a series of studies, Bigoni et al. (2008), Corso and Bigoni (2009) and Corso et al. (2018) analytically obtained the solutions of pres-stressed elasticity and showed that there is a possibility that shear bands will emerge at the stiffer tips. In addition to the line, the plane problem of rigid circular-arc inclusions was studied and the expressions of the stresses were derived in closed form (Liu and Fang, 2005). Recently, rigid line inclusion model was extended to the problem of stress concentration at small scales by introducing the surface effects (Hu and Li, 2018; Hu et al., 2021). Very recently, Zhai et al. (2021) studied the two dimension problem of a rigid circular arc inclusion reinforced decagonal quasicrystals under the remote applied tension and concentrated force. It was found that the bending moments as well as the stresses have an *r*^{(-3/2)} singularity near the tip of the rigid line and the effective shear forces exhibit an *r*^{(-5/2)} singularity, where *r* is the distance from the tip of the rigid line inclusion.

In addition to the elastic materials, inclusions in piezoelectric materials have also attracted significant attentions since those materials are very important for sensor and actuator applications (Tian et al. 2000, Wang et al. 2000a; Zhong, 2015). Piezoelectric materials reinforced by rigid line inclusions may have very important applications in future smart materials and structures. Liang et al. (1995) established the Stroh formalism for the analysis of the coupling elastic field and electric field associated with an infinite and anisotropic electroelastic material with a crack and a rigid line inclusion. Xiao et al., (2003) carried out a dislocation pileup analysis of micro-crack initiation problem of in-homogeneity tip. They found that because of the applied mechanical loading, the critical intensity factors of the stress and the electric displacement for an anti-crack can be related to the corresponding intensity factors of the normal crack, respectively. Chen et al. (2006) investigated the electro-elastic interaction of a piezoelectric screw dislocation and collinear rigid lines in a medium under anti-plane mechanical and in-plane electrical loads. They obtained explicitly the field variables and their singularities at the tips of the line inclusion and the force on the dislocation. Fang et al. (2005) studied the interaction between a generalized screw dislocation and circular-arc rigid lines at the interfaces for remote anti-plane mechanical and in-plane electric and magnetic fields in linear magnetoelectroelastic media by using the complex variable method. Later, Fang and Liu (2006) studied the electricity and elasticity interactions between a piezoelectric screw dislocation and a circular inclusion in a medium with electrically conductive rigid line at the interface.

In engineering applications, many materials are non-homogeneous composites and can be reinforced by introducing rigid line inclusions. Ballarini (1990) was the first to study a rigid line at a bi-material interface. __Jiang and Cheung (1995) __investigated the anti-plane problems of rigid line inclusions at the interface between two dissimilar materials and examined the stress state very close to the ends of the rigid line. __Kaczynski and Matysiak (2010) considered __an infinite bi-material layered system that includes an embedded rigid inclusion at the interface and examined the stress field and the effects of geometry and mechanical parameters of the dissimilar materials on the singularities of the stresses near the tips of the inclusion. Liu (2001) obtained the solution of the anti-plane elasticity problem of a periodic array of rigid line inclusions between the interfaces of the dissimilar anisotropic materials. __Lee and Kwak __(2002) determined the stress intensities for a rigid line inclusion at the bi-material interface by using the boundary element method. Prasad et al. (2005) gave the Green’s functions for a point force and interaction of dislocation with interfacial elliptical rigid inclusion in a bi-material system. They also examined the interactions between the rigid line and the interfacial inclusion. Gorbatikh et al. (2010) calculated the effect of rigid lines on the effective compliance of the composite system and the stress intensity factors at the inclusion tips.

Although extensive investigations on the rigid line inclusions have been carried out, understanding of the influence of the line inclusions on the mechanical properties of layered and composite materials is limited. The investigation of the strains and stresses near the ends of the line inclusions in a multilayered composite leads to the necessity of introducing a large number of layers. In addition, most composite materials are not isotropic in properties. Therefore, building line inclusion models for orthotropic materials is also necessary. In this paper, an analytical model for the problem of multiple inclusions in a multilayered orthotropic medium is established. The medium is treated as a system of many layers of infinite length. The material properties for each single layer are taken to be constants. The Fourier transform technique is used to obtain the general solutions of the elastic fields for each layer. The complete solution of the entire layered medium is obtained through introducing the continuity conditions of the displacements and stress across the interfaces and the mechanical boundary conditions. The solutions of strain, stress and the strain intensity factor are given. The model can be used to analyze functionally graded materials with any gradient parameter and the layered composites with any number of layers. The inclusions can be elastic and rigid. Numerical examples are given for the problems of (1) multiple parallel inclusions, (2) thermal load, and (3) multiple collinear inclusions.

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**Lee, K.Y**

**Kwak, S.G**

**Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method**

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