A unique orthogonal rotation tensor was defined and implemented into the Lawrence DYNA3D code in this paper. This tensor is vital for the implementation work of a newly formulated constitutive model proposed by M. K. to ensure the analysis was precisely integrated in the isoclinic configuration. The implementation of this unique orthogonal rotation tensor into the LLNL-DYNA3D was performed by referring to three theorems; the deformation gradient is invertible, is symmetric and positive definite and finally the rotation tensor is . The subroutine chkrot93 was adopted to check the accuracy of the proposed algorithm to calculate a proper rotation tensor . This in a matrix form as an input, and one value of output named IERR is generated. The accuracy of the proposed algorithm to define a unique orthogonal rotation tensor was tested and validated with and Plate Impact test of **orthotropic** **materials** that have so much application in real world practices. The results obtained in each material direction proved the accuracy of the rotation tensor algorithm to calculate a proper rotation tensor and provide a good agreement for **orthotropic** **materials**

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Earlier work by Butcher predicted the spall strength of AA6061-T6 should vary in accordance with the one- dimensional stress yield strength according to orientation, (Butcher, 1968). Further it was concluded that there is no significant effect on crack formation. Other works have also conducted to investigate the behaviour of spall response of **orthotropic** **materials**. Stevens and Tuler for instance concluded that the degree of pre compression, that is the shock amplitude, had no effect on the spall strength of AA6061-T6, (Stevens and Tuler, 1971). In addition, it is shown that the spall strength of AA2024-T86 decreased with increasing temperature, (Schmidt et al., 1978). However, both works concerned on geometrical effects. During shock loading, the compressive input stress is details on the rising part of the shock defined by the HEL of the material. It is observed that the shock is reflected back as a release wave when it reaches a free surface, which consequently takes the material back to ambient stress conditions. Releases from the rear of flyer and target can be arranged by controlling the thickness of the specimen and flyer plate to meet in the middle of the target itself, and where they do so, a zone of net tension will result. If that tension exceeds the tensile strength of the material, the net result is failure (spall), which can be detected by appropriate measurement techniques, (Vignjevic et al., 2002).

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Much research has been carried out in respect to complex **materials** behaviour of **orthotropic** **materials** under dynamic loading conditions, leading to results in various technologies involving analytical, experimental and computational methods. Despite of this current status, it is generally agreed that there is a need for improved constitutive formulation as well as the corresponding procedures to identify the parameters for these models. Modelling finite strain deformation and failure in such **materials** requires an appropriate mathematical description. This is a real deal since an appropriate formulation can be very complex, specifically to deal with the orientations of **materials** orthotropy (Sitnikova et al., 2015). Moreover, there are numerous mechanics of **materials** issues that have yet to be solved, related to **orthotropic** elastic and plastic behaviour. For example, shape changes resulting from a deformation process on a continuum level can be very complex when dealing with **orthotropic** **materials** since the co-linearity of the principal axis of the stress and strain tensors is no longer in place. Based on this motivation, this research project is conducted to develop a constitutive formulation to predict the behaviour of commercial aluminium alloys undergoing large deformation including failure.

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incremental and experimental approaches [12- 18]. But, no inverse finite element method has been developed for design initial blank of **orthotropic** **materials** in deep drawing process. In this work, a developed inverse finite element method was presented to obtain strain and stress distribution in final shape and design initial blank of deep drawing process for **orthotropic** sheets. Accuracy of the present method was evaluated by comparison with the results of ABAQUS, experimental results and numerical methods.

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Припускається, що гранична величина енергії викривлення форми, яка описується рівнянням 15, є сталою при всіх комбінаціях σ x , σ y та σ z де застосовується дев’ять пружних сталих матері[r]

However, in the case of **orthotropic** **materials** this co-linearity is not in place. Hence the equivalent relationship cannot be deﬁned for **orthotropic** **materials**. If one maintains the assumption that pressure is the state of stress induced by an isotropic state of strain (uniform compression or expansion) then a more general deﬁnition of pressure is required, [73]. This leads to a number of possible de ﬁ nitions of pressure as a vector in the principal stress space which is not co-linear with the conventional hydrostatic alignment for **orthotropic** **materials**. To explore this statement further Vignjevic has proposed a new expression for generalized pressure or stress related to uniform compression. The ability to describe shock propagation in **orthotropic** **materials** is investigated with experimental plate impact data and showed a good agreement with the physical behaviour of the consid- ered material (carbon ﬁ bre reinforced epoxy).

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[3,4], studied the stresses in anisotropic non -homogeneous cylinders and sphere. Add-Alla, et al [5], studied the effect of non-homogeneity on the composite infinite cylinder of isotropic material. Add-Alla, and Abo-Dahb [6], investigated time-harmonic sources in a generalized magneto -thermo- viscoelastic continuum with and without energy dissipation. The rotation of a non-homogeneous composite infinite cylinder of **orthotropic** and isotropic material, and the thermal stresses in a rotating non-homogeneous **orthotropic** hollow cylinder has been investigated by El-Naggar, et al. [7-9]. In [10], Zhang and Hasebe investigated the elasticity solution for a radially non-homogeneous hollow cylinder. Ding, et al. [11, 12, 13, 14], studied the elastodynamic solution of a non - homogeneous **orthotropic** hollow cylinder, and a solution of a non-homogeneous **orthotropic** cylindrical shell for a xy - symmetric plane strain dynamic thermoelastic problem. Haojiang, et al. [15], studied the elas todynamic solution of a non-homogeneous **orthotropic** hollow cylinder. Oral, et al [16], investigated the effect of radially varying moduli on stress distribution of non-homogeneous anisotropic cylindrical bodies. Grigorenko,et al [17], studied the equilibrium of elastic hollow inhomogeneous cylinders of corrugated elliptic cross-section. Abd-Alla, et al. [18,19], investigated the effect of the non-homogeneity on the composite infinite cylinder of isotropic and **orthotropic** **materials**. Theotokoglou, et al. [20], investigated the radially non-homogeneous elastic ax- symmetric problem. Ersalan [21], studied the elastic-plastic deformations of rotating variable thickness annular disks with free pressurized and radial constrained boundary conditions. Zenkour [22], studied stress distribution in rotating composite structures of functionally graded solid disks. Lesan [23], investigated the deformation of inhomogeneous **orthotropic** elastic cylinders. Some problems of elasticity theory for anisotropic bodies of cylindrical form studied by Grigorenko, et al. [24],

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m) caused by the mentioned load for the gearless drivetrain. As it happened for the steel structure, the highest deformation was observed to take place at the edges of the rim. In the case of the stator, the same disc sub- structures were applied. Nevertheless, the geometry of the stator is completely different to that of the rotor. The configuration with the two discs helps to make the overall structure stiffer and therefore reduce the deflection. But as composites are **orthotropic** **materials**, the fibre

Due to low density and low coefficient of thermal expansion, **orthotropic** **materials**, namely, the polymer matrix composites are finding their way in many of the engineering application. The present study investigates the performance of the **orthotropic** annular fin. The thermal conductivity along radial and axial direction is governed by the thermal conductivity ratio (K), which is an important parameter for **orthotropic** annular fin. The work then numerically examines in two dimensions, the thermal performance of the annular fin at different thermal conductivity ratio (K) for the rectangular profile using the finite difference method. Different convective heat transfer coefficient are considered at the fin sides and the fin tip. The effect of various parameters such as axial Biot number (Bi s ), contact Biot number (Bi c ), tip Biot number (Bi e ), aspect ratio (b/a), thermal conductivity ratio (K)

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Srivastava et al. [22] solved the problem of interaction of shear waves with Griﬃth crack situated at the interface of two bonded dissimilar elastic half spaces. The problem becomes more diﬃcult and complicated when boundaries are present in the media. Li [14] obtained the analytical solution for a static problem of two bonded **orthotropic** strips containing an interfacial crack. The dynamical problem was studied by Matbuly [18] and obtained the analytical expression of stress intensity factor. Diﬀraction problems involving multiple cracks had been studied by many authors. But most of the problems were either involving diﬀraction of shear waves or diﬀraction in inﬁnite media. E. Lira-Vergara and C. Rubio-Gonzalez [12-13] obtained the dynamic stress intensity factor of interfacial ﬁnite cracks in **orthotropic** **materials**. Itou [7] considered the diﬀraction problem of an antiplane shear wave by two coplanar Griﬃth cracks in an inﬁnite elastic medium. Stress distribution near periodic cracks at the interface of two bonded dissimilar **orthotropic** half planes was studied by Garg [6]. The problem of diﬀraction of elastic waves by two coplanar Griﬃth cracks in an inﬁnite elastic medium was solved by Jain and Kanwal [10]. The transient response of two cracks at the interface of a layered half space had been investigated by Kundu [11]. Mandal and Ghosh [15] solved the problem of interaction of elastic waves with a periodic array of coplanar Griﬃth cracks in an **orthotropic** medium. Diﬀraction problem of three coplanar Griﬃth cracks in an **orthotropic** medium was considered by Sarkar et al. [20]. Das [3] and others considered the problems containing a Griﬃth crack in an transversly **orthotropic** medium. Mukherjee et al. [17] have studied the interaction of three interfacial Griﬃth cracks between bonded dissimilar **orthotropic** half planes and ﬁnd out the stress intensity factor. Satapathy et al. [19] considered an **orthotropic** strip containing a Griﬃth crack which is ﬁnally solved. Das et al. [4]are ﬁnding the Stress intensity factor of an edge crack in bonded **orthotropic** **materials**. Dynamic stress intensity factors of multiple cracks in an **orthotropic** strip with FGM coating was studied by Monfared and Ayatollahi [16]. Sinharoy [21] solved elastostatic problem of an inﬁnite row of parallel cracks in an **orthotropic** medium under general loading. Itou [9] solved the problem of dynamic stress intensity factors of three collinear cracks in an **orthotropic** plate subjected to time-harmonic disturbance

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Since the offset of strain typically used to define the yield point (stress at which material begins to deform plastically) of metals is very small; approximately 0.2% of the strain, the formulation is limited to small elastic deformation with large rotation; hence, the Mandel stress tensor is symmetric. Hill’s yield criterion is adopted to characterise the plasticity of **orthotropic** **materials**. It can be observed that this yield criterion has been widely used in industrial simulations and provides reasonably good results as well as being numerically efficient. The expansion of the new material model yield surface (hardening part) is characterised by a referential curve of the thermally micromechanical-based model, the Mechanical Threshold Model (MTS). Generally, a finite “saturation stress”, or what can be described as a constant but small hardening rate, is achievable at finite strains of most metallic **materials**. The other models such as Johnson-Cook (JC) and Zerilli-Armstrong (ZA) models are unable to capture this behaviour. Moreover, the MTS model fits the experimental data more reasonably at various strains of the saturation stress.

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Following various methods, the elastic elds of various loadings, inclusion and inhomogene- ity problems, and interaction energy of point defects and dislocation arrangement have been discussed extensively in the past. Generally, all **materials** have elastic anisotropic properties which mean the mechanical behavior of an engineering material is character- ized by the direction dependence. However, the three dimensional study for an anisotropic material is much more complicated to obtain than the isotropic one, owing to the large number of elastic constants involved in the calculation. In particular, transversely isotropic and **orthotropic** **materials**, which may not be distinguished from each other in plane strain and plane stress, have been more regularly studied. A brief look at the literature on micropolar **orthotropic** continua shows that Iesan [7]-[9] analyzed the static problems of plane micropolar strain of a homogeneous and **orthotropic** elastic solid, the torsion prob- lem of homogeneous and **orthotropic** cylinders in the linear theory of micropolar elasticity and bending of **orthotropic** micropolar elastic beams by terminal couple. Nakamura et al. [21] applied the nite element method to **orthotropic** micropolar elasticity. Kumar and Choudhary [10]-[14] have discussed various problems in **orthotropic** micropolar continua. Singh and Kumar [26] and Singh [27] have also studied the plane waves in micropolar generalized thermoelastic solid.

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The proposed formulation to define the isoclinic configuration adopted in [11] has been briefly explained in the previous publication [12]. Therefore, this paper is constructed to rigorously discuss the formulation of the proposed orthogonal rotation tensor and further explore the capability to deal with finite strain deformation in three-dimensional mode. The proposed formulation in [11] that is developed in the isoclinic configuration ̅ provides a unique treatment for elastic and plastic anisotropy. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a Mandel stress tensor combined with the new stress tensor decomposition generalized for **orthotropic** **materials** [13]. The formulation of new Mandel stress tensor ̂ defined in the isoclinic configuration is given by

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Abstract. The analytic function of complex variable including a material parameter is analyzed fully. Typical model II crack is considered to **orthotropic** **materials**. By constructing new stress function, the mechanic analysis for crack-tip singular stress field is carried out. Boundary problems are studied and the formulae for stress fields are derived.

In the time-dependent case the situation is much less investigated and here we only mention the nonlinear identification of a temperature-dependent **orthotropic** material, [14], the space-dependent anisotropic case considered in [11], and the recovery of the leading coefficients of a heterogeneous **orthotropic** medium, [1, 5].

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ABSTRACT: Composite **materials** have a long history of usage. Straw was used by the Israelites to strengthen mud bricks. Plywood was used by the ancient Egyptians to achieve superior strength arid resistance to thermal expansion as well as to swelling. Now days, fiber-reinforced, resin-matrix composite **materials** that have high strength-to-weight and stiffness-to-weight ratios have become important in weight-sensitive applications such as aircraft and space vehicles. The key material properties for usual engineering applications are strength and stiffness. The fibers are stiff and have high strength and they are expected to carry the load to which the structure is submitted. The matrix has low strength and low stiffness and it gives the shape to the component and transfers the loads to fibers and between them. The usual design criterion for composite material is based on trying to align the fibers with most critically loaded directions of mechanical component. This paper based on an investigation to find out elastic constants and other mechanical properties of an **orthotropic** (glass epoxy) composite lamina theoretically and by finite element analysis because composite structure may not yield the exact results due to **orthotropic** or non linear behavior of composite material. KEY WORDS: Glass epoxy, ANSYS, Fiber Volume Fraction, Modulus of Elasticity

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The paper is devoted to dynamic design of thick **orthotropic** cantilever plates by ap- plying the bimoment theory of plates, which takes into account the forces, moments and bimoments; and the theory takes into account nonlinear law of displacements distribution in cross section of the plate. The methods for constructing bimoment theory are based on Hooke’s Law, three-dimensional equations of the theory of dy- namic elasticity and the method of displacements expansion into Maclaurin series. The article gives the expressions to determine the forces, moments and bimoments. Bimoment theory of plates is described by two unrelated two-dimensional systems with nine equations in each. On each edge of the plate, depending on the type of fas- tening, nine boundary conditions are given. As an example, the solution of the prob- lem of dynamic bending of thick isotropic and **orthotropic** plate under the influence of transverse dynamic loads in the form of the Heaviside function is given. The equa- tions of motion of the plate are solved by numerical method of finite differences. The numerical results are obtained for isotropic and **orthotropic** plate. The graphs of changes of displacements and stresses of faces surfaces of the plate are presented. Maximum values of these displacements are found and analyzed. It is shown that by Timoshenko theory numerical values of stresses are much smaller compared to the ones obtained by bimoment theory of plates. Maximum numerical values of genera- lized displacements, forces, moments, and bimoments are obtained and presented in tabular form. The analysis of numerical results is done and the conclusions are drawn.

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The composite layer was tested in three directions to obtain its **orthotropic** properties such as elasticity, shear modulus and Poison’s ratio. Damage parameters were determined for warp and weft direction and were updated later using high and low fiber fraction specimens cut out from the FML specimens. Based on the observations made in this work, the following conclusions may be derived:

The parameters of the **orthotropic** model can be obtained by static stiffness test or modal test. Two blocks attached by a kind of glue are shown in Fig. 6. The blocks are considered rigid, and the mass block 1 is fixed on the ground. For mass block 2, the mass is m, the inertia around the z-axis is J z , and

A Simplified Approach to the Propagation of Edge Waves in a Thinly Layered Laminated plate with Stress Couples Under Biaxlal Initial Stresses, Acta Geoph.. Wave Propagation in a Thinly T[r]

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