Three-dimensional Constitutive Model Research Articles

Validation of a three-dimensional finite element constitutive modeling approach for a thermoplastic polyurethane calibrated with uniaxial tests

This paper presents the three-dimensional finite element constitutive modeling validation for a Thermoplastic Poly-Urethane (TPU) material. The material parameters are those given in a previous study obtained by calibrating the constitutive model with uniaxial material-level test results (1-D tensile and compression tests). Confined compression tests were performed to estimate the bulk modulus for more accurate material characterization. The parameters were validated by comparing the results of experimental tests on TPU components with those obtained by its equivalent finite element simulations. The TPU component tests comprised cyclic compression tests of (i) a TPU cylinder, (ii) a solid 100 mm diameter TPU ball, and (iii) a 100 mm diameter TPU ball with an 80 mm steel core. In all tests, the constitutive model parameters showed an excellent performance in representing the mechanical behavior of the material observed in the tests, including the nonlinear stiffness and hysteresis. Finally, a brief case study is presented to illustrate the applicability of the validated constitutive model parameters in modeling a novel TPU damper for structural control before its manufacturing and experimental testing. The validated constitutive modeling approach and available material parameters will allow the performance of reliable and early-stage finite element simulations to prototype the mechanical behavior of TPU components of different shapes and boundary conditions and thus gain relevant insight into the component level response before manufacturing and testing.

Nonlinear mechanical behaviour and visco-hyperelastic constitutive description of isotropic-genesis, polydomain liquid crystal elastomers at high strain rates

The mechanical behaviour of isotropic-genesis, polydomain liquid crystal elastomers (I-PLCEs) at various strain rates is systematically investigated via experiments, theoretical analysis, and numerical modelling. Experiments encompassing SEM (scanning electron microscope), DSC (differential scanning calorimetry), TGA (thermogravimetric analyser), quasi-static and dynamic (SHPB – split Hopkinson pressure bar) mechanical tests, as well as drop-weight impact tests, are undertaken to identify the nonlinear, large-strain, rate-dependent relationship between compressive stress and deformation of the I-PLCEs studied. Subsequently, a three-dimensional compressible visco-hyperelastic constitutive model for the material is established based on the summation of Cauchy stress components. The as-used model yields good agreement with experimental data, particularly an excellent description of the mechanical responses at high strain rates of 103∼104 s−1. The fully-calibrated constitutive model is implemented in the commercial finite element code ABAQUS via a virtual user-defined material (VUMAT) subroutine. The inhomogeneous deformation processes of the I-PLCEs, corresponding to impact by a hemispherically-tipped drop weight, which induces complex stress states, are also well described. Finally, when evaluated by two dimensionless physical parameters, the I-PLCEs demonstrate a more pronounced strain rate sensitivity in terms of dynamic strength and impact toughness compared to other commonly used materials, highlighting their superior performance in dynamic loading scenarios. The present study is helpful for the design and development of impact-resistant LCE-based materials and structures.

A 3D finite deformation constitutive model for anisotropic shape memory polymer composites integrating viscoelasticity and phase transition concept

The phase transition theory of shape memory polymers (SMPs) often involves a phenomenological assumption that the reference configuration of the newly transformed phase deviates from that of the initial phase. This distinction serves as a crucial mechanism in the manifestation of the shape memory effect. However, elucidating the precise definition of the reference configuration of the transformed phase poses a significant challenge in the formulation of the constitutive model. To tackle this challenge, a three-dimensional (3D) finite deformation constitutive model incorporating effective phase evolution for SMPs has been developed. This model merges insights from the classical viscoelastic framework with the phase transition theory. The anisotropic thermo-viscoelastic constitutive model is further developed by introducing hyperelastic fibers, which integrate the anisotropy of the fibers into a continuous thermodynamic framework through structure tensors. Implemented within the ABAQUS software via a user material (UMAT) subroutine, the proposed model has been meticulously validated against experimental data, showcasing its prowess in simulating stress-strain responses and shape memory characteristics of SMPs and their composites (SMPCs). This innovative model stands as an invaluable instrument for the design and of sophisticated SMP and SMPC structures.

Interaction and compatibility between steel and concrete of circular CFST stub columns with high-strength materials

To promote the application of high-strength-materials in concrete-filled steel tubular (CFST) columns, finite element modeling, and parametric analysis were conducted on circular CFST stub columns filled with normal-strength concrete (NSC) and ultra-high strength concrete (UHSC). For simulating properties of concrete under passive confinement, a three-dimensional constitutive model for NSC and UHSC was established, of which the stress-strain relation and evolution of dilation angle are both related to confining pressure and axial plastic strain. The model was verified by tests of CFHSST (concrete-filled high-strength steel tube) and UHSCFST (ultra-high-strength concrete-filled steel tube), demonstrating that the interaction between concrete and steel tube can be revealed correctly. Parameter analysis of CFST stub columns filled with NSC and UHSC was carried out, with the grade of steel tube covering from Q235 to Q890 and the D/t ratio ranging from 8.9 to 75. With the decrease of D/t ratio, the deformation capacity and post-peak performance of columns are improved. A recommended range of steel contribution ratio of UHSCFST was proposed for the consideration of cost and safety, which is 25% ∼ 63%. Based on the analysis of interaction between concrete and steel tubes, the compatibility of the two materials is recommended, of which the yield of steel tube occurs after contact with concrete and before the concrete reaches its peak strength. Besides, the formula in EC4 overestimates the load-bearing capacity of CFST stub columns with high-strength materials. Therefore, a modified formula was proposed that reduces the enhancement factor of concrete strength and enables a safer prediction of load-bearing capacity.

Triaxial-mechanical and permeability properties of coal body under varying CO2 pressures

The sequestration of CO2 in coal seams has become an effective way to curb greenhouse-gas emissions. Coal mechanics and permeability properties are key factors affecting the safe sequestration of CO2 in coal seams, and both are significantly affected by the CO2 injection pressure. In this study, triaxial compression-permeability tests were conducted on coal under varying CO2 and limited confining pressures using a measurement system to determine the coupled mechanical properties and adsorption permeability of coal. The effects of CO2 pressure on the mechanical properties and evolution of coal permeability were investigated. A three-dimensional statistical damage constitutive model of coal that considers CO2 adsorption damage was established. The results showed that the stress–strain curve of the coal was divided into four stages. During the first two stages, the amplitudes of the permeability changes were small, whereas at the peak stress point, a permeability “jump” phenomenon occurred, after which an increase in stress resulted in a slower amplitude increase in permeability. The CO2 pressure had an evident damage-deterioration effect on the mechanical properties of the coal samples, and the primary failure characteristic was shearing damage. The higher the CO2 pressure, the higher the degree of internal fragmentation and fracture network complexity of the coal body. During the triaxial compression-permeability experiment, the normalized permeability of the coal samples tended to increase slowly, followed by a rapid increase and back to a slow increase. However, with increasing CO2 pressure, the initial permeability of the coal samples decreased, whereas the normalized permeability increased sharply. The rationality and accuracy of the constitutive model were verified by comparing the established constitutive model and test stress–strain curves of the coal samples. The results of this study can provide theoretical references for CO2 geological storage and facilitate the selection of appropriate CO2 injection pressures.

Modeling the heterogeneous and anisotropic plastic deformation of lath martensite

The plastic behavior of microscale lath martensite samples is highly anisotropic. Depending on the orientation, the deformation of such samples may be heterogeneous, with only a few localized slip traces, while the remainder of the sample remains largely elastic. Although several continuum plasticity models that account for the anisotropy exist, they cannot reproduce the heterogeneous response observed in experiments. In this study, a model for lath martensite at the microscale is proposed which captures the orientation-dependent heterogeneous behavior observed in experiments. Before formulating the model we first study in detail two idealized cases, in which two different deformation mechanisms are activated. In both cases, the lath martensite is modeled using a discrete slip plane model. In the model, the activation stress of the individual slip systems varies randomly in space according to a distribution based on the underlying dislocation motion. The two configurations differ only in the orientation of the applied tensile load relative to that of the laths — either perpendicular or at 45°. In the latter case, slip along the so-called habit plane results in localized plastic deformation, while the former results in a more diffuse activation of plasticity. Insights obtained based on the idealized cases are used to formulate a three-dimensional constitutive model which captures both deformation mechanisms. The model is applied to microtensile tests of single-packet lath martensite samples. It is shown that the orientation-dependent heterogeneity is accurately captured by the two deformation mechanisms accounted for by the model.

A three-dimensional elastoplastic constitutive model incorporating Lode angle dependence

A comprehensive three-dimensional elastoplastic constitutive model is presented to characterize the stress-strain behavior of cement stone under the true triaxial stress state. This constitutive model incorporates a three-dimensional yield function and a three-dimensional potential function. The three-dimensional yield function is designed to accurately represent the true triaxial stress state during hardening. The three-dimensional potential function is devised to depict the plastic flow direction under true triaxial stress state. The yield and potential functions include parameters that control the shape of the deviatoric and meridian planes, and these parameters vary with the plastic internal variable. Consequently, the yield function can accurately describe the stress state, and the potential function can precisely capture the variations in plastic flow direction. Additionally, a detailed procedure for determining the parameters of the yield function and potential function is proposed based on the full deformation process. The constitutive model is presented in the form of analytical solution. The comparison of experimental data with the constitutive model confirms its accuracy and validity. A sensitivity analysis of the deviatoric and meridian parameters in the potential function is performed, shedding light on their impact on the model behavior. Furthermore, the significance of incorporating Lode angle dependence into the potential function is discussed, emphasizing its essential role in accurately capturing strain in the direction of the intermediate principal stress.

Experimental and numerical analysis of bending performance for a glass-epoxy composite with integrated sheet-shaped shape memory alloy actuator elements

Shape memory alloys (SMA) belong to the group of smart materials and show particularly a promising potential for an active functionalization of lightweight composite structures with respect to enabling smart structures. In this work, various manufacturing concepts for the integration of thin sheet-shaped SMA actuator elements in a thin glass-epoxy composite beam are experimentally and numerically studied with regard to its bending deformation performance realized by thermal induced shape memory effect of the SMA actuator element. For numerical modeling, the shape memory effect of the SMA on structural level, a three-dimensional thermomechanical constitutive material model is applied. In the experiments, the same deformation pattern of the composite beam was observed for all integration concepts investigated, with the degree of deformation showing strong differences both when the shape memory effect is activated and deactivated. The complex material behavior of the SMA actuator under bending deformation is modeled via the finite element method with sufficient accuracy including existing tension and compression loads.

Linear elastic diatomic multilattices: Three-dimensional constitutive modeling and solutions of the shift vector equation

Diatomic multilattices are congruences of simple lattices each made out of atoms of two possible chemical species. We here constitutively characterize, in three dimensions, diatomic multilattices for the geometrically and materially linear elastic regime. We give the most generic expression for the energy for ( n + 1 ) diatomic multilattices and characterize explicitly the tensors present in such an expression for all 122 two-color point groups. For the specific case of diatomic 2- and 3-lattices, we delineate how one can solve the shift vector equation in the static case. For cases where the unique solution of the shift vector in terms of the strain tensor is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem.

Performance evaluation of the Generalized Bounding Surface Model in the simulation of Cajicá clay subjected to monotonic loading

Different constitutive models, based on principles of mechanics and experimental evidence, have been developed over several decades to represent and predict the stress-strain behavior of soils subjected to various loading conditions. The Generalized Bounding Surface Model (GBSM) is one of them and is defined as a fully three-dimensional elastoplastic constitutive model for saturated cohesive soils that employ the bounding surface plasticity in conjunction with a nonassociative flow rule and a soil microfabric-inspired rotational hardening rule. The GBSM has been successfully validated on numerous laboratory reconstituted soils, but not on natural or undisturbed soil samples. Hence, in this paper the predictive capabilities of the GBSM are evaluated in the simulation of the monotonic behavior of an undisturbed cohesive soil called “Cajicá clay” from the high plain of Bogotá in Colombia. In the first instance, an isotropic consolidation test and a set of axisymmetric triaxial compression and extension tests are conducted using an automated triaxial equipment, to experimentally describe the response of the undisturbed soil. From experimental data, the parameters associated with the GBSM model are calibrated to finally evaluate its capabilities in the simulation of a cohesive natural soil. A comparison between experimental data and numerical simulations is presented to show both performance and advantages of the GBSM model.

Elastic-viscoplastic model for coarse-grained soil considering particle breakage

This paper presents a novel elastic-viscoplastic constitutive model for reproducing the time-dependent behaviour of coarse-grained soil considering particle breakage, by integrating the Unified Hardening (UH) model, the elastic-viscoplastic (EVP) model and the overstress theory. The relationship between the degree of particle breakage and the loading rate is established, and the state variables associated with the critical state of coarse-grained soil are derived to jointly consider both time and particle breakage. A new three-dimensional elastic-viscoplastic constitutive model is then constructed through combining the one-dimensional viscoplastic hardening parameter with a secondary consolidation coefficient considering particle breakage. The proposed model requires 19 parameters and it can well describe the influence of time-dependency and particle breakage to the shear, dilatancy, and compression behaviours of coarse-grained soil with various confining pressures or initial void ratios. Comparisons between the model predictions and experimental data are employed to validate the capability of the proposed model to replicate the time-dependent behaviour of coarse-grained soil.

On the modeling of cavitation and yielding in rubber-toughened amorphous polymers: Fully implicit implementation and optimization-based calibration

In the present contribution, a finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is proposed to predict the nonlinear response of neat, porous and rubber-toughened amorphous polymers. In this context, a well-known nucleation law is modified and coupled with a nucleation criterion to suppress nucleation under compression and shear loadings, as well as to identify its material parameters directly. The growth of both the nucleated and initial voids is ruled by a modified version of the Gurson’s potential, ensuring the coherence in the energy dissipation between the macro and microscales. The post-yield strain hardening of the material is modeled with the well-established eight-chain model. A fully implicit integration algorithm is formulated under the infinitesimal strains functional format, and both constitutive state update and consistent tangent modulus are properly established. A three-stage optimization-based calibration procedure is designed to identify the model’s material parameters efficiently. This decoupled calibration strategy is only possible due to the well-defined influence of the model’s material parameters in the numerical response. The predictive capability of the proposed constitutive model is validated against literature results for polycarbonate and three rubber-filled polycarbonate blends. Moreover, the numerical response of the model is assessed under different triaxial states and is further compared with the homogenized response of a Representative Volume Element of the voided-amorphous polymer microstructure. The results highlight that the model can capture the typical behavior of amorphous polymers, namely yield stress, strain-softening and strain hardening under different loading conditions, as well as the decrease of stress associated with the increase of the void volume fraction. The additional numerical analyses conducted, validated some constitutive assumptions. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated.

Formulation and implementation of a hypoplastic constitutive model for interface behavior of mechanical joints

The effects of the contact interface must be considered during the design process of the assembly structures connected by mechanical joints. An advanced three-dimensional (3D) constitutive model is needed to describe the complex behavior of the contact interface in a detailed finite element (FE) model. However, the simplified Coulomb friction law, which is a bilinear model without considering the microslip behavior and stiffness degradation, is often applied to the surface-surface contact element for modeling the interface in commercial FE programs. In this paper, a new 3D hypoplastic constitutive model for the nonlinear interface behavior is presented. This model is built on the framework of hypoplasticity developed by Kolymbas and the hypothesis of continuous media. First, the general hypoplastic model is derived based on the kernel formula and several restrictions and simplified according to the characteristics of the contact interface. Then, the coefficients of the model are associated with the widely used physical parameters by combining the model with the initial stiffness ratio, normal contact law and Coulomb law. Benefiting from the hypoplastic theorem, the deduced constitutive model has a simple mathematical formulation in which only five parameters are contained. Simulations of several monotonic and cyclic shear tests show that the model can represent the salient characteristics of the joint interface (e.g. stick/microslip/macroslip behaviors, motion coupling and pressure-dependency). Thus, it can be used to predict the response of the joint interface from arbitrary multi-axial load histories more precisely. The constitutive model is then implemented in a commercial FE program via analysis of a double-beam structure subjected to harmonic excitation. And vibration experiments on several preload levels are performed to validate the effectiveness and the applicability of the proposed model to the real joints. The results calculated from the FE model with the improved reduction techniques are in good agreement with the experimental data.

Study on cross-scale temperature effect and damage failure evolution constitutive model of phthalazine ether sulfone ketone

Poly(phthalazine ether sulfone ketone) (PPESK) is a new type of thermoplastic resin materials with high temperature resistance and corrosion resistance. However, the failure evolution regularities of PPESK at different temperatures still remain poorly understood, which greatly limits the application of this composite. Therefore, quasi-static tensile/bending tests and microscanning microscopic characterization were used to study the multi-scale failure constitutive equation of PPESK. Based on cross-scale experimental observation methods, the spatial damage morphology of materials at different temperatures is explored, and the temperature boundary effect of PPESK is proposed to reveal the failure and failure mechanisms of materials in a wide temperature range. The initiation and evolution of pores at the micro scale of PPESK and the cumulative damage caused by craze propagation and cracking are essential factors of material failure, which also exhibit strong temperature sensitivity. The specific manifestation is the high-temperature viscous flow effect of the resin at the micro scale, as well as the gradual evolution trend of quasi-brittleness and viscoelastic-plasticity between the 295 K–472 K temperature range in the macro performance, which increases its plastic deformation ability by 1–5 times. Specifically, the mechanism of cross-scale micro-pore initiation and crack propagation was introduced, and a cross-scale, fully three-dimensional failure damage evolution constitutive model of PPESK was constructed for the first time in a wide temperature range. The secondary development of the material constitutive model in the finite element software was carried out, and the construction of the constitutive equation of the PPESK cross-scale failure was completed. Finally, the mapping relationship between the material's crack failure form, mechanical behavior, and the constitutive model was further verified by experiments. It is established that the proposed PPESK multi-scale failure constitutive model enables one to effectively describe the complex mechanical response and failure law of thermoplastic resin materials under temperature gradients. Moreover, this theoretical tool is promising for the design of independently controllable high-temperature resistant thermoplastic resin materials, thereby providing important guidance means for engineering applications.

A constitutive model for amorphous thermoplastics from low to high strain rates: Formulation and computational aspects

In this paper, a recently proposed finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is extended to predict the nonlinear response of amorphous polymers from low to high strain rates. The model accounts for the influence of distinct molecular mechanisms, which become active at different deformation rates. Therefore, the constitutive equations include two relaxation phenomena to describe the strain rate sensitivity of amorphous polymers. Well-established rheological elements are adopted to define visco-elasticity (generalized Maxwell elements) and visco-plasticity (Eyring dashpots). In addition, strain hardening is modeled with a plasticity-induced (nonlinear) hardening element which is extended to distinguish between the contribution of the two transitions. From a computational viewpoint, a fully implicit integration algorithm is derived, and a highly efficient implementation is obtained. It is shown that it is possible to reduce the return mapping system of equations to only two independent (scalar) nonlinear equations. A four-stage optimization-based calibration procedure is proposed to identify the model’s material parameters in a completely unsupervised way. The predictive capability of the constitutive model is validated against literature results for polycarbonate and poly(methyl methacrylate), accounting for temperature and strain rate dependencies under different loading conditions. The results show that the model can capture the transition in the yield behavior and predict the post-yield large strain behavior over a wide range of strain rates. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated. Despite the adiabatic conditions observed under high strain rates, the model replicates the associated effect of temperature through strain rate dependency.

A creep model for ultra-deep salt rock considering thermal-mechanical damage under triaxial stress conditions

To investigate the specific creep behavior of ultra-deep buried salt during oil and gas exploitation, a set of triaxial creep experiments was conducted at elevated temperatures with constant axial pressure and unloading confining pressure conditions. Experimental results show that the salt sample deforms more significantly with the increase of applied temperature and deviatoric loading. The accelerated creep phase is not occurring until the applied temperature reaches 130 °C, and higher temperature is beneficial to the occurrence of accelerated creep. To describe the specific creep behavior, a novel three-dimensional (3D) creep constitutive model is developed that incorporates the thermal and mechanical variables into mechanical elements. Subsequently, the standard particle swarm optimization (SPSO) method is adopted to fit the experimental data, and the sensibility of key model parameters is analyzed to further illustrate the model function. As a result, the model can accurately predict the creep behavior of salt under the coupled thermo-mechanical effect in deep-buried condition. Based on the research results, the creep mechanical behavior of wellbore shrinkage is predicted in deep drilling projects crossing salt layer, which has practical implications for deep rock mechanics problems.

Domain Switching-Based Nonlinear Coupling Response for Giant Magnetostrictive Materials.

This paper proposes a multilevel three-dimensional constitutive model based on a microscopically phenomenological approach from the domain rotation mechanism, which is a fully coupled self-consistent homogenization scheme considering the interactions between elastic-inelastic strain and hysteresis. Considering the interactions among magnetic domains, grains, polycrystalline complexes, and macroscopic phenomenology, we predict the nonlinear magnetostrictive response of Terfenol-D under different types of external force loads and magnetic excitations in various thermal environments involving multi-fields of coupled magnetic, elastic, thermal, and mechanical phenomena. The average values of the mechanical bulk strains for different magnetization states are obtained at the grain scale utilizing Boltzmann functions and a self-consistent homogenization scheme. A Taylor series expansion of the Gibbs function concerning the field variables and an adapted Jiles-Atherton model are used to construct the hysteresis coupled constitutive relations at the macroscopic scale. The results associated with the experiments show that the established model can reasonably predict the magnetostrictive response under different external mixed stimuli. It can provide theoretical guidance for the precise control of nonlinear vibrations and the optimal design of the rotating giant magnetostrictive transducers at both microscopic and macroscopic multiple scales.

A multiphysical computational model of myocardial growth adopted to human pathological ventricular remodelling

We present a novel three-dimensional constitutive model that describes an electro-visco-elastic-growth response on the myocardium with a fully implicit staggered solution procedure for the strong electromechanical coupling. The novel formulations of the myocardium allows us to simulate and analyze the remodelling of actively contracting human ventricular heart models which consist of growing viscoelastic myocardium where the growth direction is determined based on its mechanical state at each time step. The total deformation gradient is multiplicatively decomposed into a mechanical-active part and a growth part, where the mechanical-active part is further split into elastic, viscous, and active components. Unconditional stability of time integration is ensured by a backward Euler integration scheme. With the developed model, the myocardium can experience stretch-driven longitudinal (fibre) growth and stress-driven transverse (cross-fibre) growth. To validate the developed approach, two simulations regarding pathological ventricular remodelling are implemented: two divergent types of remodelling of a left ventricular model driven by hemodynamic overloads and ventricular remodelling triggered by acute myocardial ischemia in a biventricular heart model.

Infrared radiation constitutive model of sandstone during loading fracture

The significant increase in energy demand will inevitably boost the exploitation of natural resources. The exploitation of resources including coal, oil, geothermal, natural gas, shale gas, and other energy has to be extracted with deep mining to fulfill the industry energy demand. With the increase of mining depth and the utilization of deep rock mass, the problem of high ground stress has become increasingly prominent. During high ground stress conditions, the mechanical behavior of hard rock will change significantly. Constructing a reasonable stress–strain damage constitutive model of hard rock is the basis for the design of a resource mining construction scheme, numerical simulation analysis, more accurate acquisition of rock mechanical characteristics, and evaluation of engineering rock stability. This is very crucial that needs to be addressed in a better way to solve the scientific problems of rock engineering. In this research, the infrared radiation observation experiments of sandstone biaxial under different lateral stresses are carried out. It is found that the infrared radiation energy has a near power function relationship with the effective stress in loading and fracture process. The quantitative characterization method of infrared radiation of the first invariant of stress and the second invariant of deviator stress is established. The cumulative high-temperature point scale factor amplitude is proposed to characterize the plastic volumetric strain of sandstone. Based on the effective stress and plastic strain, the plastic and damage models are established respectively. The three-dimensional plastic damage constitutive model of loaded sandstone based on infrared radiation is constructed. The model has clear input parameters having physical significance, and the compaction stage of sandstone is considered. The stress prediction of sandstone during biaxial loading is realized through the secondary development of a finite element software subroutine. The research results can lay a theoretical foundation for the analysis and evaluation of the mechanical characteristics of hard rock in the process of deep energy exploitation.

A stabilized hybrid peridynamic method compatible with constitutive models of different dimensions

Peridynamics (PD) has been increasingly widely employed to address cracking and fracture behaviors of engineering materials. However, the existing two types of PD, namely, bond based peridynamics (BPD) and state based peridynamics (SPD), have their respective limitations. The BPD is practical and capable of solving real engineering problems, but cannot work with three-dimensional (3D) constitutive models. The SPD is able to use 3D constitutive models but suffers from the so called “zero-energy mode” problem especially when the system is highly nonlinear. This paper presents a stabilized hybrid PD method (HPD) that can be effectively compatible with 1D to 3D constitutive models and does not have zero-energy mode issue for highly nonlinear problems. The proposed HPD creates bonds between each PD point and other PD points inside its horizon, and obtains the total force on the PD point by all bond forces. For 2D or 3D (or simply written as nD) constitutive models, each bond force consists of an axial force and a shear force evaluated at the midpoint of the bond, and is calculated by the nD stress tensor multiplied by the direction and area of the bond. The nD stress can be obtained by using an nD constitutive model based on the nD strain at the midpoint of the two connected PD points. The nD strain is estimated by using displacements of PD points within a new sub-horizon, which is defined by the overlap of two horizons of the connected PD points. For 1D constitutive model cases, the axial force is obtained by the axial strain and the shear force is ignored, and the HPD is degraded to BPD. Four application examples are presented to verify the proposed HPD method, i.e., static analyses for a linear cantilever beam and a nonlinear soil column using 2D material models, seismic analysis for a soil column using 3D nonlinear model and dynamic analysis of a plate with pre-crack using 1D and 3D models. The responses (e.g., displacements, stresses, strains at representative points) are analyzed and compared with those by FEM, BPD and SPD. The HPD method presented herein is demonstrated to be accurate, stable, and compatible with various material models of different dimensions to solve a wide range of complicated problems.