## What is/are Isotropic Material?

Isotropic Material - Thin-walled corrugated structures have been widely used in engineering applications for centuries, because corrugation enables engineers to tailor directional dependent properties despite the structures being made of isotropic materials.^{[1]}For the description of yielding, an isotropic yield criterion which allows to differentiate between isotropic materials was used.

^{[2]}Conventional techniques, suitable for the typical design of the strips made from anisotropic material such as steel, are not useful for СFRP strips.

^{[3]}Almost all earlier mechanical analysis of implants presumed bone as an isotropic material while the bone is anisotropic.

^{[4]}We introduce an FFT-based method to compute the effective crack energy of heterogeneous, locally anisotropic materials.

^{[5]}However, 3DCP has many challenges such as competing rheological requirements, weak interlayer bonding, difficulty in integrating reinforcement, and anisotropic material behaviour.

^{[6]}1 Technical challenge: earing prediction for an anisotropic material Most sheet forming benchmarks include information such as: the hardening law parameters, based on limited experimental data (generally, r-values, yield stresses and ultimate strengths for 3 loading directions) extracted from uniaxial tension tests the coefficients for certain yield functions, available in the material libraries of most commercial software.

^{[7]}However, high f r are obtainable only along the easy axis direction of the magnetic anisotropic materials.

^{[8]}Consequently, the initial systems of governing equations for vibration analysis of sandwich structures made of isotropic materials are derived.

^{[9]}Its dual-responsiveness allows the independent control of different DoFs for complex shape-morphing behaviors with anisotropic material properties.

^{[10]}17 at 808 nm), superior to most 2D anisotropic materials.

^{[11]}Future composite aircraft wing designs will exploit anisotropic material properties by aeroelastic tailoring and include active control methods for manoeuvre and gust load alleviation.

^{[12]}This paper presents a 3D model of a terahertz photoconductive antenna (PCA) using black phosphorus, an emerging 2D anisotropic material, as the semiconductor layer.

^{[13]}Using the 3D microstructure of a real cathode, the stresses inside a free-standing electrode and model cells with a thin and a thick LLZO separator are calculated for the charging cycle considering isotropic and anisotropic material properties of LCO as well as non-textured and textured crystallographic alignment.

^{[14]}Once corrected stresses of homogenous layers satisfy von Mises criterion, Ramberg-Osgood curve is used to update elastic constants of isotropic materials.

^{[15]}By extending the conventional scattering canceling theory, we propose a new design method for thermal cloaks based on isotropic materials.

^{[16]}In addition, we demonstrate how the use of anisotropic materials enables us to set this optimum state at smaller wavelengths, even close to ultraviolet (UV), as compared to isotropic dielectrics, and we reveal the key role of the incident field's polarization in achieving optimized operation for both prolate and oblate configurations.

^{[17]}In this study, finite-element simulations of seven shear test geometries were evaluated for an isotropic material in a series of virtual experiments by varying the input hardening response.

^{[18]}This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton.

^{[19]}The isotropic and anisotropic materials with damping effects are also considered.

^{[20]}In this work, a technique of how to detect the presence of a crack-like defect and estimate its orientation in an anisotropic material by means of two orthogonal shear waves is presented.

^{[21]}All bones and external fixators were assigned with isotropic material properties while the cartilages were simulated with hyper-elastic.

^{[22]}In this work, the virtual crack closure-integral technique is implemented to a mixed finite element, in addition with the stiffness derivative procedure, to evaluate the energy release rate of crack extension in anisotropic materials.

^{[23]}The ductile fracture behavior of anisotropic materials was investigated and modeled by the uncoupled ductile fracture criterion for aluminum alloys 6016-AC200.

^{[24]}The stress state of hollow cylinders with oval cross-section made of orthotropic and isotropic materials is analyzed using spatial problem statement and analytical methods of separation of variables, approximation of functions by discrete Fourier series, and numerical discrete-orthogonalization method.

^{[25]}The main focus of this work is to predict the effective thermal conductivity of anisotropic materials based on the three-dimensional reconstruction of their fibrous structure, obtained from X-ray micro-tomography.

^{[26]}The stress field is also obtained from Hooke’s law for isotropic materials.

^{[27]}The proposed PIL method has the following advantages: (a) it can be applied to isotropic/anisotropic plate structures; (b) it doesn’t require any signal interpretation, making it attractive for active impact monitoring systems; (c) for anisotropic materials it doesn’t require the accurate knowledge of the wave velocity in all directions of propagation; (d) it doesn’t need a reference database; and (e) it remains effective even in the presence of noise.

^{[28]}A graphene layer, with isotropic surface conductivity of σ, has been sandwiched between two adjacent anisotropic materials.

^{[29]}They require only a low-to-moderate amount of training data and training time to learn without human guidance the constitutive behavior also of complex nonlinear and anisotropic materials.

^{[30]}All columns were made of two layers of isotropic materials characterised by various mechanical properties and were simple supported at the ends.

^{[31]}Another key element of this question is that initially, isotropic material can lose this property due to a cold pre-straining process and the existence of the Bauschinger effect.

^{[32]}In addition, this work proposes a novel anisotropic material interpolation scheme, which integrates both matrix and fiber design variables (both with material nonlinearity) into the stored-energy function.

^{[33]}The relationship between the mechanical properties of anisotropic materials and their thermophysical characteristics was experimentally revealed using the example of Scots pine ( Pınus sylvéstri s L) wood.

^{[34]}Besides, a multi-material interpolation scheme based on the Porous Anisotropic Material with Penalization (PAMP) is used for topology optimization.

^{[35]}The results show that the structures of these perovskite derivatives are stable and they are all anisotropic materials.

^{[36]}The method enables the determination of gas flow (in each flow direction) in microchannels forming an orthogonal network, characteristic of isotropic materials.

^{[37]}In the study of wave propagation through plates and laminates, there are no complete three-dimensional solutions especially if the plate or plies are made of an anisotropic material.

^{[38]}Our results also show that anisotropic material properties of either a tissue or an embedded nodule render the embedded tumor nodule undetectable using indentation.

^{[39]}Unlike all the other codes that are presently available, the software presented here is capable of simulating both isotropic and anisotropic materials comprised of single or multiple domains.

^{[40]}Rock is generally regarded as a heterogeneous and anisotropic material containing massive initial defects, such as cracks, joints, and porosities.

^{[41]}Tensile tests for different sheet orientations are conducted both experimentally and numerically to adjust the anisotropic material parameters by inverse parameter identification for aluminium EN AW-6014 and steel HCT590X.

^{[42]}Refined calculation formulas for determining the one - and two-axis stress state of an anisotropic material, taking into account the anisotropy of the thermoacoustic coefficients of transverse waves, are proposed.

^{[43]}It is assumed that the object is made of isotropic materials with equal values of permittivity and permeability and consists of a spherical volume of material with a positive spatially uniform refractive index and an adjacent spherical layer of material with a negative inhomogeneous refractive index (i.

^{[44]}The objective is to determine both the forces and paths of cracks propagating in elastoplastic and anisotropic materials.

^{[45]}This suggests that the incorporation of collagen is an efficient way to supplement the lack of confinement while reinforcing mechanical stability to the highly anisotropic materials.

^{[46]}The isotropic material 4-node bilinear plane strain elements (CPE4R) were used to mesh the solid cylinder cross-section.

^{[47]}This paper deals with the possible field of application of ultrasonic Surface Reflection Method (SRM) to achieve the mechanical characteristics of isotropic materials.

^{[48]}Transition metal ions doped PbI2 is strongly anisotropic material having high molecular weight, high polarizability and strong spin-orbit coupling.

^{[49]}Indeed, recurrent anisotropic material properties and fracture strength still remains as the key challenges to be tackled in FFF, so as to get close to mechanical performances obtained by injection molding.

^{[50]}

## finite element method

In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty.^{[1]}The formula for the model I fracture toughness of the transversely isotropic material is obtained on the basis of the finite element method (FEM) together with the J-integral.

^{[2]}The superiority of the proposed method over Monte Carlo, solid isotropic material with penalization (SIMP) and polynomial chaos expansion (PCE) using classical finite element method (FEM) is demonstrated via two practical examples with compliances in material uncertainty and loading uncertainty improved by approximately 11% and 10%, respectively.

^{[3]}In this paper, a method is proposed for extracting fracture parameters in isotropic material cracking via a stable generalized/extended finite element method.

^{[4]}In the present analysis, ground test conditions and forced vibration test are simulated using 3D finite element method to check the effect of number of pads and shims on the flex seal, to study the effect of isotropic material and composite material on the flex seal for shims.

^{[5]}The method is established by combining the finite element method (FEM) with two different optimization procedures called bi-directional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP), respectively.

^{[6]}We utilize the finite-element method with unstructured tetrahedral meshes for the spatial discretization supporting irregular survey geometries and anisotropic material parameters.

^{[7]}In the present investigation, free vibration analysis of thin isotropic materials of bonded metallic plates under various boundary conditions is found using finite element method.

^{[8]}The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method.

^{[9]}

## topology optimization method

This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method.^{[1]}The work provides an exhaustive comparison of some representative families of topology optimization methods for 3D structural optimization, such as the Solid Isotropic Material with Penalization (SIMP), the Level-set, the Bidirectional Evolutionary Structural Optimization (BESO), and the Variational Topology Optimization (VARTOP) methods.

^{[2]}This paper presents a multicomponent topology optimization method for designing structures assembled from additively manufactured components, considering anisotropic material behavior for each component due to its build orientation, distinct material behavior, and stress constraints at component interfaces (i.

^{[3]}Most traditional topology optimization methods have been developed for isotropic materials and adapted to consider direction-dependent materials only to define the optimum shape based on the fixed properties.

^{[4]}Combining the isomorphic mapping matrix with the solid isotropic material with the penalization topology optimization method, the topological model of the 6-DOF spatial compliant mechanism is constructed, and a topological structure of the 6-DOF spatial compliant mechanism is derived which has the same differential kinematic characteristics as the Gough–Stewart prototype platform.

^{[5]}

## Solid Isotropic Material

1007/s00158-020-02545-z ), that is based on DBNs and the solid isotropic material with penalization (SIMP) approach.^{[1]}The solid isotropic material with penalization method is employed to obtain the material distribution.

^{[2]}This paper presents an approach based on the topology optimization, the Solid Isotropic Material with Penalization (SIMP) method.

^{[3]}The method of moving asymptotes (MMA) is used for optimization, and material interpolation is handled with solid isotropic material with penalization (SIMP).

^{[4]}In this paper a multiple SIMP (solid isotropic material with penalization model) of variable density method is proposed to solve the problem of muli-material topology optimization.

^{[5]}The Solid Isotropic Material with Penalty (SIMP) method is utilized to define the material properties as a function of the design variables.

^{[6]}The density filter solid isotropic material with penalization method combined with threshold projection is developed.

^{[7]}In engineering practice, a given space is first described using the finite element method and, subsequently, density-based method with solid isotropic material with penalty.

^{[8]}Based on the solid isotropic material with penalization (SIMP) interpolation model, the mathematical statement of the proposed optimization design is formulated by integrating the transient objective function over the time interval that considers thermal dissipation energy minimization.

^{[9]}In the first phase, the topology scheme for the gripper is proposed via the solids isotropic material with penalization method in terms of a full consideration of stress constraint and equal forces of both hands.

^{[10]}This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method.

^{[11]}Therein, the Heaviside penalization of the discrete material optimization method and the solid isotropic material with penalization scheme are separately employed to optimize the fiber orientation and the layout of the damping material.

^{[12]}A lightweight design process and method for the compression molding of automotive interior parts and a mathematical model for the optimization of the solid isotropic material penalty (SIMP) (power law) material interpolation of the concave and convex molds are established.

^{[13]}Using the variable density method, the dynamic topology optimization model of a long-span continuum structure is built based on the density interpolation model of a solid isotropic material with penalization (SIMP).

^{[14]}The traditional solid isotropic material with penalization model is modified to eliminate the artificial localized modes.

^{[15]}This formulation couples positional finite elements to the solid isotropic material with penalization method.

^{[16]}Second, using the simulation’s results as input, topology optimization of the support structures by applying the Solid Isotropic Material with Penalization (SIMP) method is executed.

^{[17]}A penalization scheme, SIMP (Solid Isotropic Material with Penalization) method is used to determine the optimum distribution of material, and void has been incorporated in the initial design.

^{[18]}The solid isotropic material with penalization (SIMP) and parametrized level set methods are compared for the temperature-constrained topology optimization.

^{[19]}Optimization of the structural adhesives and spotwelds was carried-out using SIMP (Solid Isotropic Material with Penalization) based topology optimization.

^{[20]}The work provides an exhaustive comparison of some representative families of topology optimization methods for 3D structural optimization, such as the Solid Isotropic Material with Penalization (SIMP), the Level-set, the Bidirectional Evolutionary Structural Optimization (BESO), and the Variational Topology Optimization (VARTOP) methods.

^{[21]}The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization.

^{[22]}At the same time, the Solid Isotropic Material with Penalization (SIMP) interpolation is applied to model the microstructural topology on the microscale.

^{[23]}The optimization model was built on the basis of the solid isotropic material penalization method with the introduction of additional restrictions in the model of searching for pseudo-densities of the material, taking into account the duration of the force action on the stamp under multicyclic loading.

^{[24]}This article focuses on the topology optimization for additive manufacturing considering self-supporting constraint based on the Solid Isotropic Material with Penalization (SIMP) framework.

^{[25]}Two classical compliance minimization problems are solved within this paper and benchmarked against a Solid Isotropic Material with Penalization (SIMP)–based topology optimization.

^{[26]}In order to improve the manufacturability of topology optimization results, this paper proposes a hybrid method based on explicit description of Moving Morphable Components (MMC) and implicit description of Solid Isotropic Material with Penalization (SIMP).

^{[27]}Two schemes of material interpolation within the framework of the solid isotropic material with penalization (SIMP), i.

^{[28]}In addition, the solid isotropic material with penalty scheme (SIMP) is used to optimize the layout of the damping material.

^{[29]}The superiority of the proposed method over Monte Carlo, solid isotropic material with penalization (SIMP) and polynomial chaos expansion (PCE) using classical finite element method (FEM) is demonstrated via two practical examples with compliances in material uncertainty and loading uncertainty improved by approximately 11% and 10%, respectively.

^{[30]}However, high resolution is desired for optimum structures, but it normally leads to a computationally intractable puzzle, especially for the solid isotropic material with penalization (SIMP) method.

^{[31]}The method is established by combining the finite element method (FEM) with two different optimization procedures called bi-directional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP), respectively.

^{[32]}Combined with a numerical analysis based on finite elements, the SIMP (Solid Isotropic Material with Penalization) method formulation, which is defined with the criterion of minimum strain energy restricted by the volumetric fraction, is used for the development of the models with the ABAQUS® v.

^{[33]}Both are considered by simulation of the build process and defined as constraints in the context of a Solid Isotropic Material with Penalization method based topological optimization.

^{[34]}In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field and, on the other hand, on the well-known Solid Isotropic Material with Penalization (SIMP) approach.

^{[35]}This interpolation scheme, unlike the traditional Solid Isotropic Material Penalization (SIMP) interpolation, is observed to perform better in terms of approximating the structure’s load-bearing capacity, primarily due to its formulation on the lattice’s stiffness matrices.

^{[36]}We adopt the Bi-value Coding Parameterization (BCP) scheme combined with the Solid Isotropic Material with Penalization (SIMP) method to interpolate the stored energy functions.

^{[37]}Solid isotropic material with penalization method is selected to update the design variables.

^{[38]}Taking the relative density of the finite element of the constrained damping layer as design variable, the solid isotropic material with penalization method is used to realize the optimal topological distribution of the damping material on the surface of the metal substrate.

^{[39]}Macro-scale topology optimization is performed using the traditional solid isotropic material with penalization method, while the unit structures composing the macro-structure can have various shapes to improve the heat conduction performance according to the simultaneous optimization process.

^{[40]}Combining the isomorphic mapping matrix with the solid isotropic material with the penalization topology optimization method, the topological model of the 6-DOF spatial compliant mechanism is constructed, and a topological structure of the 6-DOF spatial compliant mechanism is derived which has the same differential kinematic characteristics as the Gough–Stewart prototype platform.

^{[41]}Thus, we propose an approach based on the SIMP (Solid Isotropic Material with Penalization) method and the adjoint method in order to solve the HET TO design problem.

^{[42]}In this study, a hybrid method for density-related topology optimization is proposed, which consists of two parts: the discrete level-set method (LSM) based on solid isotropic material with penaliz.

^{[43]}This paper presents a hybrid algorithm for topology optimization of lightweight cellular materials and structures simultaneously by combining solid isotropic material with penalization (SIMP) and bi-directional evolutionary structural optimization (BESO).

^{[44]}The TOM implemented is based on the solid isotropic material with penalization (SIMP), the dynamic adjoint sensitivity, and on the optimization solver known as sequential linear programming (SLP).

^{[45]}

## Transversely Isotropic Material

It indicates that the MAE behaves as a transversely isotropic material in the presence of an external magnetic field.^{[1]}Exact solution of axisymmetric wave propagation problem in radially and functionally graded circular cylinder made from combination of isotropic and transversely isotropic materials is obtained.

^{[2]}Combining the mechanic theory of transversely isotropic material, the anisotropy parameters of the tight sandstone are analyzed, as well as the influence on the hydrofracturing technology for tight sandstone in the field.

^{[3]}In the second part application to effective elastic coefficients of transversely isotropic materials such as clay rocks, in the frame of homogenization theory is presented to illustrate the impact of concavity parameter on overall properties.

^{[4]}In this paper, the nonlocal elasticity theory is applied to study the propagation of plane wave and Rayleigh-type surface wave in an incompressible, rotating and transversely isotropic material.

^{[5]}The formula for the model I fracture toughness of the transversely isotropic material is obtained on the basis of the finite element method (FEM) together with the J-integral.

^{[6]}On the other hand, while studying composite cellular materials, the anisotropic property of each of the carbonfiber-reinforced composite members is considered by postulating a transversely isotropic material property relation between the components of the incremental second Piola-Kirchhoff stress tensor and the incremental Green-Lagrange strain tensor.

^{[7]}The analysis of the results demonstrated that the purely elastic transversely isotropic material model is adequate for predicting behavior, at least before nonlinearities occur.

^{[8]}Results show that the material anisotropy of transversely isotropic materials exerts a strong influence on the stress intensity factors.

^{[9]}On the other hand, a Zener model and a model using complex constants are extended to model the transversely isotropic material.

^{[10]}In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled based on the strain energy density function proposed based on a novel framework.

^{[11]}The three-dimensional solution is derived in this article for the double-coated structure of the transversely isotropic material.

^{[12]}For consistency with the infinitesimal theory, it is well known that there are three necessary conditions on the derivatives of W $W$ (evaluated in the undeformed state) that have to be to be satisfied in terms of the three independent elastic moduli of the linear theory for incompressible transversely isotropic materials.

^{[13]}Also, the lateral and transverse elastic moduli of layers were found to be approximately the same, and therefore the constitutive behaviour of the layers can be treated as transversely isotropic material.

^{[14]}This article presents a short review of the harmonic general solutions for uncoupled elasticity of transversely isotropic materials with thermal and other effects.

^{[15]}

## Homogeneou Isotropic Material

While geometry, mass and stiffness can often be characterised quite accurately, at least for homogeneous isotropic materials, the experimental quantification of structural damping is a time consuming endeavour.^{[1]}A new peridynamic bond failure model is proposed for mixed-mode crack fracture analysis in material interface and homogeneous isotropic materials, which utilize bond failure criteria presented for mixed-mode peridynamic bonds using the angle-dependent formations of critical stretch (CS) or critical energy density (CED).

^{[2]}When a rock slope of significant height involves predominantly one type of jointed rock mass, the scale of the problem is frequently such that the rock mass (intact rock and rock discontinuities) can be regarded as a homogeneous isotropic material.

^{[3]}For homogeneous isotropic materials, the stiffness matrixes for RBSM are equal to that for ISEM, and there are only four items different in the stiffness matrix for AEM.

^{[4]}The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth.

^{[5]}Based on the principle of superposition and an equivalent indentation method to solve an axisymmetric external crack problem, a series of closed-form solutions are derived for power-law punch profiles which reduce to the existing solutions for homogeneous isotropic materials and for paraboloidal geometries as special cases.

^{[6]}The focus is on the practically relevant class of thin plates (with thickness h ) made of a homogeneous isotropic material that is perforated with periodic distributions (with unit-cell size e ) of monodisperse holes spanning a large range of porosities, from the dilute limit to nearly the percolation threshold.

^{[7]}

## Elastic Isotropic Material

Constitutive relations of two classes are proposed for nonlinear elastic isotropic materials, which, in case of purely volumetric deformation, are reduced to the Murnaghan’s equation of state.^{[1]}The problem lies in the fact that when one is solving applied problems of ice engineering, ice is often considered as an elastic isotropic material, and its stress-strain state (SSS) is studied in terms of the theory of bending of elastic plates.

^{[2]}A linear elastic isotropic material model for the solidified polymer was used to obtain the solid foam properties.

^{[3]}

## Transverse Isotropic Material

We derive closed-form solutions for reverberant elastography in anisotropic elastic media by adapting the framework used in electromagnetic theory to treat transverse isotropic materials.^{[1]}However, when it comes to transverse isotropic material, this approach has a natural limitation due to the isotropic radial stresses; particular attention to the boundary conditions and the proper design of pressurization steps is warranted.

^{[2]}The closed form solution is corroborated through numerical non-linear fracture analyses for different geometries with isotropic and transverse isotropic material properties.

^{[3]}

## Nonlinear Isotropic Material

It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law.^{[1]}Inverse numerical simulation was used to simulate the indentation tests to determine and verify the parameters of a nonlinear isotropic material model for the weldment of LBW.

^{[2]}

## Virtual Isotropic Material

This paper attempts to validate the application of the Virtual Isotropic Material Concept (VIMC) in combination with the average strain energy density (ASED) criterion to predict the critical load in notched laminated composites.^{[1]}A newly proposed concept, called the Virtual Isotropic Material Concept (VIMC), accompanying with two well-known brittle failure models, namely the maximum tangential stress (MTS) and the mean stress (MS) criteria, is utilized for the first time for theoretical LPF load prediction under mixed mode I/II loading.

^{[2]}

## Thin Isotropic Material

Therefore, in the present work a 3D hexahedral solid-shell element, based on the initial work of Schwarze and Reese [2,3], which has shown promising results for the forming of thin isotropic materials [1], is extended for highly anisotropic materials.^{[1]}In the present investigation, free vibration analysis of thin isotropic materials of bonded metallic plates under various boundary conditions is found using finite element method.

^{[2]}

## Conventional Isotropic Material

Conventional joining elements like rivets and screws or simple clamping are designed for an application in conventional isotropic materials such as steel or aluminum.^{[1]}Three types of material are tested: conventional isotropic materials (like XPS), compressible anisotropic materials (like wood fiber insulation) and heterogeneous anisotropic materials (like light-earth biobased concrete).

^{[2]}

## isotropic material property

Its dual-responsiveness allows the independent control of different DoFs for complex shape-morphing behaviors with anisotropic material properties.^{[1]}Future composite aircraft wing designs will exploit anisotropic material properties by aeroelastic tailoring and include active control methods for manoeuvre and gust load alleviation.

^{[2]}Using the 3D microstructure of a real cathode, the stresses inside a free-standing electrode and model cells with a thin and a thick LLZO separator are calculated for the charging cycle considering isotropic and anisotropic material properties of LCO as well as non-textured and textured crystallographic alignment.

^{[3]}All bones and external fixators were assigned with isotropic material properties while the cartilages were simulated with hyper-elastic.

^{[4]}Our results also show that anisotropic material properties of either a tissue or an embedded nodule render the embedded tumor nodule undetectable using indentation.

^{[5]}It has high reproducibility which is desirable for high volume of parts, it produces isotropic material properties and where necessary can be used to provide unique alloys not possible from wrought material.

^{[6]}Indeed, recurrent anisotropic material properties and fracture strength still remains as the key challenges to be tackled in FFF, so as to get close to mechanical performances obtained by injection molding.

^{[7]}The basic equation of elastic mechanics has been established according to the anisotropic material properties.

^{[8]}The shear and rotary inertia effects are neglected by considering a slender-shaped beam with homogeneous and isotropic material properties.

^{[9]}It consists of a numerical model to capture the residual stress due to the tool–part interaction and a viscoelastic model considering the anisotropic material properties.

^{[10]}On the other hand, while studying composite cellular materials, the anisotropic property of each of the carbonfiber-reinforced composite members is considered by postulating a transversely isotropic material property relation between the components of the incremental second Piola-Kirchhoff stress tensor and the incremental Green-Lagrange strain tensor.

^{[11]}Because of the anisotropic material properties, high demands on the tool performance and process stability are set.

^{[12]}Our model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties.

^{[13]}The surface patterns are rotated at different angles and changed into different shapes to change the anisotropic material properties of the PDMS specimens.

^{[14]}Although thin-walled tubes can exhibit anisotropy, such numerical models have traditionally included isotropic material properties.

^{[15]}However, previous theories describing their mechanical behavior rarely consider multilayer and anisotropic material properties, resulting in limited application and significant analysis errors.

^{[16]}Herein, the influence of linear-elastic, local anisotropic material properties as well as residual stresses resulting from the compression molding of LFT on the stiffness-optimized design of beaded plates is investigated.

^{[17]}The present research contributes in the structural topology optimization with the manufacturing limitation called the overhang constraint and the optimization of the anisotropic material properties.

^{[18]}, anisotropic material properties, signal forms and parametrizations); and (b) simulating these systems at global scale with high-accuracy requires a large computational cost, often requiring days or weeks on a supercomputer.

^{[19]}Numerical models that consider the cortical tables and the diploë as domains with separate isotropic material properties are constructed for each bone segment using a routine that identifies the cortical table-diploë boundaries from micro-computed tomography scan images, and reconstructs a three-dimensional geometry layer by layer.

^{[20]}Computational models show that the RVE can be used to extract homogenized anisotropic material properties of full cells.

^{[21]}Linearly elastic, inhomogeneous, and isotropic material properties were assigned to bone based on density distributions reconstructed from the medical images.

^{[22]}The closed form solution is corroborated through numerical non-linear fracture analyses for different geometries with isotropic and transverse isotropic material properties.

^{[23]}Due to their anisotropic material property, chatter and poor surface quality are easy to occur during machining.

^{[24]}However, applied to machining of carbon fibre reinforced polymers (CFRP), the process complexity potentially increases due to the anisotropic material properties, the elastic spring back potential of the material, and the distinct mechanical wear due to the highly abrasive carbon fibres.

^{[25]}Fused filament fabrication (FFF) introduces anisotropic material properties to the final parts, which includes minimum interlayer values.

^{[26]}This arrangement results in anisotropic material properties, which depend on local fiber orientations.

^{[27]}

## isotropic material model

This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton.^{[1]}Due to the unique processing conditions in PBF-LB/M, materials often develop a dominating microstructure that leads to anisotropic mechanical properties, and thus isotropic material models fail to account for the orientation-dependent mechanical properties.

^{[2]}The analysis of the results demonstrated that the purely elastic transversely isotropic material model is adequate for predicting behavior, at least before nonlinearities occur.

^{[3]}Continuum predictions using five different isotropic material models are compared head-to-head with molecular dynamics (MD) predictions for a 50 nm cylindrical pore in β-HMX subject to a range of shock strengths.

^{[4]}Here the research question was answered if an anisotropic material model better describe the grinding process and process forces compared to an isotropic material model.

^{[5]}An anisotropic material model according to type 36 (MAT_036 3-PARAMETER_BARAT) was applied.

^{[6]}Most of the existing ultrasonic guided wave tomography approaches to map structural changes in plate-like waveguides are based on the assumption of an isotropic material model.

^{[7]}These two tests have been modelled using a cohesive zone formulation for the separating interface and a hyperelastic anisotropic material model via an implicit static analysis.

^{[8]}Inverse numerical simulation was used to simulate the indentation tests to determine and verify the parameters of a nonlinear isotropic material model for the weldment of LBW.

^{[9]}A linear elastic isotropic material model for the solidified polymer was used to obtain the solid foam properties.

^{[10]}

## isotropic material behavior

This paper presents a multicomponent topology optimization method for designing structures assembled from additively manufactured components, considering anisotropic material behavior for each component due to its build orientation, distinct material behavior, and stress constraints at component interfaces (i.^{[1]}The heterogeneous, nonlinear, anisotropic material behavior of biological tissues makes precise definition of an accurate constitutive model difficult.

^{[2]}The optimized model accurately describes the anisotropic material behavior observed in tensile and stress relaxation tests in a wide range of temperature, specimen orientation, and strain rate.

^{[3]}This paper presents a modified Mayergoyz-based vector hysteresis model to describe the anisotropic material behavior of nonoriented (NO) steels over a wide range of rotational excitations.

^{[4]}Additively manufactured materials possess process-related structural characteristics, such as residual porosity or an anisotropic material behavior, thereby leading to distinguished properties of the substrate/coating system compared to conventionally fabricated substrate materials.

^{[5]}The experimental results revealed anisotropic material behavior for flexural, tensile, shear and compressive loading.

^{[6]}

## isotropic material concept

This paper attempts to validate the application of the Virtual Isotropic Material Concept (VIMC) in combination with the average strain energy density (ASED) criterion to predict the critical load in notched laminated composites.^{[1]}A newly proposed concept, called the Virtual Isotropic Material Concept (VIMC), accompanying with two well-known brittle failure models, namely the maximum tangential stress (MTS) and the mean stress (MS) criteria, is utilized for the first time for theoretical LPF load prediction under mixed mode I/II loading.

^{[2]}

## isotropic material parameter

Tensile tests for different sheet orientations are conducted both experimentally and numerically to adjust the anisotropic material parameters by inverse parameter identification for aluminium EN AW-6014 and steel HCT590X.^{[1]}We utilize the finite-element method with unstructured tetrahedral meshes for the spatial discretization supporting irregular survey geometries and anisotropic material parameters.

^{[2]}

## isotropic material characteristic

In this research, mechanical characterisation procedures that experimentally evaluate the anisotropic material characteristics of a fibre-reinforced printable concrete (FRPC) are presented.^{[1]}We calculated 3 × 3 Mueller matrix elements, which can be used to described an intuitive overview of the anisotropic material characteristics.

^{[2]}

## isotropic material penalization

The optimization model was built on the basis of the solid isotropic material penalization method with the introduction of additional restrictions in the model of searching for pseudo-densities of the material, taking into account the duration of the force action on the stamp under multicyclic loading.^{[1]}This interpolation scheme, unlike the traditional Solid Isotropic Material Penalization (SIMP) interpolation, is observed to perform better in terms of approximating the structure’s load-bearing capacity, primarily due to its formulation on the lattice’s stiffness matrices.

^{[2]}