Strain Energy Density Function Research Articles

Controllable Deformations of Unconstrained Ideal Nematic Elastomers

We establish that, for ideal unconstrained uniaxial nematic elastomers described by a homogeneous isotropic strain-energy density function, the only smooth deformations that can be controlled by the application of surface tractions only and are universal in the sense that they are independent of the strain-energy density are those for which the deformation gradient is constant and the liquid crystal director is either aligned uniformly or oriented randomly in Cartesian coordinates. This result generalizes the classical Ericksen’s theorem for nonlinear homogeneous isotropic hyperelastic materials. While Ericksen’s theorem is directly applicable to liquid crystal elastomers in an isotropic phase where the director is oriented randomly, in a nematic phase, the constitutive strain-energy density must account also for the liquid crystal orientation which leads to significant differences in the analysis compared to the purely elastic counterpart.

Data-driven hyperelasticity, Part II: A canonical framework for anisotropic soft biological tissues

In this work, we present a novel anisotropic data-driven hyperelasticity framework for the constitutive modeling of soft biological tissues that allows direct incorporation of experimental data into the constitutive model, without requirement of a predetermined mathematical formula for the strain–energy density function. The data-driven framework is constructed through a dispersion-type anisotropic formulation based on a generalized structure tensor in the sense of Holzapfel et al. (2015) that take into account in- and out of plane dispersion. The partial derivatives of the strain energy density functions are replaced with appropriate B-spline interpolations where the control points are calibrated against experimental data obtained from uniaxial tension, triaxial shear, and (equi)biaxial tension deformations. The model calibration phase incorporates the normalization condition and the polyconvexity condition is enforced through the control points of the B-splines in order to ensure a stable constitutive response that allows unique solution in finite element analysis. The predictive capabilities of the proposed model are shown against linea alba, rectus sheath, aneurysmal abdominal aorta, and myocardium tissues. On the numerical side, the stress and moduli expressions of the model are derived and implemented into the finite element method. The performance of the model is demonstrated through representative boundary value problems.

Remarks on bifurcation of an inflated and extended swellable isotropic tube

We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.

Numerical Analysis and Validation of Characterization of Polydimethylsiloxane Using Hyper-elastic Constitutive Models

The most researched elastomer in recent years is polydimethylsiloxane (PDMS), which has several uses in various engineering industries. One of the PDMS’s key characteristics is its hyper-elasticity nature, which enables the production of sensors, flexible electrical circuits, transducers, and antennas. This study used the hyper-elastic constitutive models to predict the mechanical behavior of incompressible, isotropic, and hyper-elastic material PDMS under uniaxial tension. These models are curve-fitting tools that consist of strain energy density and stress functions. To pursue the analysis, a new formulation of PDMS substrate was proposed, and a tensile test was performed to evaluate its stress-strain behavior. The experimental data was implemented on various hyper-elastic models using Abaqus, like Mooney-Rivlin, Yeoh, Ogden, and reduced polynomial models. The goodness of fit of every model was evaluated by calculating R2 values. Consequently, among these models, the reduced polynomial model with 6 material constants possessed the highest R2 value (0.9936) and was considered the best-fit model among the other models. Furthermore, the material constants of this model were applied to the 3D dumbbell-shaped model of PDMS in Abaqus for its validation. The boundary conditions were applied on the model similar to the experimental setup, as 33 mm displacement on one end and the other was fixed with all DOF. For mesh quality and mesh sensitivity of the material, various mesh sizes with the linear formulation (C3D8RH) were utilized, and the best mesh size was selected to evaluate very close results with the experimental.

New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.

An experimentally validated anisotropic visco-hyperelastic constitutive model for knitted fabric-reinforced elastomer composites

A new anisotropic visco-hyperelastic constitutive model has been developed to predict the time-dependent deformation response of flexible, knitted fabric-reinforced elastomer composites. These composites exhibit a time-dependent response due to the viscoelastic behavior of the elastomer, while the anisotropic reinforcement effect is attributed to the knitted fabrics. The proposed modeling approach features an invariant-based strain energy density function that incorporates the architecture of knitted fabrics and the hyper-viscoelastic behavior of elastomers. In this study, the elastomer is considered as a viscous neo-Hookean material, and the knitted fabrics are modeled as cords with negligible stiffness in bending. The proposed material model is implemented in commercial Finite Element Analysis (FEA) software, Abaqus (version 2017), through a user-defined material subroutine, UMAT. This approach enables the determination of the constitutive behavior of the composite based on the constituent elastomer, fiber properties, and fabric architectures. The proposed model is verified through a discrete 3D FEA model, in which the elastomer and fabric reinforcement are modeled explicitly. The approach is further validated through physical testing including uniaxial tension, torsion, and three-point bending. Since the proposed strain energy-based constitutive model can predict the deformation response of knitted fabric-reinforced elastomer composites based on the fabric architecture, it can be employed as an efficient modeling tool to aid the design of knitting pattern for target mechanical properties.

Probabilistic fatigue evaluation of notched specimens considering size effect under multiaxial loading

Fatigue evaluation of notch components is key to ensuring the structural integrity of grand equipment and implementing fatigue resistant design, while fatigue models that effectively couple multiaxial stress condition, notch and size effects are still lacking. Accordingly, a new probabilistic fatigue model by combining the weakest link theory with critical plane theory is proposed in this work. Firstly, the critical plane location is determined by calculating the stationary point of the relevant strain energy density function. Then, a new effective stress concept is proposed to establish the connection between experimental data of smooth specimens and equivalent stress in the critical damage region of notched specimens. Meanwhile, a critical damage region considering the size effect is defined to determine the effective stress. Finally, the experimental data of EN8 medium carbon steel and Q345 alloy steel are used for model validation and comparison and the results show that the model has a higher accuracy than the other four models. Additionally, the curves P−σeff‾−Nf can effectively describe the fatigue life scatter.

Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.

Data-driven hyperelasticity, Part I: A canonical isotropic formulation for rubberlike materials

Data-driven hyperelasticity shows great promise for modeling the mechanical response of rubberlike materials. It enables an automated linkage between experimental data and mechanical response without a priori knowledge of specific analytical expression for the strain energy density function or the stress expression. In this study, we propose a new data-driven approach with three distinct kinematic approaches; (i) invariant-based formulation, (ii) modified invariant-based approach, and (iii) principal stretch-based formulation to model the hyperelastic response of rubberlike materials. Within this context, we replace the partial derivatives of the strain energy density functions with appropriate B-spline interpolations, using a set of control points to implement various multiaxial loading scenarios, such as uniaxial tension, pure shear, and equibiaxial tension deformations. To ensure a polyconvex and stable constitutive response, we enforce the polyconvexity requirement into the B-spline interpolation along with appropriate normalization conditions. In order to obtain the control points of the B-splines, we train the proposed data-driven approach with respect to the Treloar and Kawabata datasets. On the numerical side, the stress and moduli expressions are derived for the finite element implementation. The predictive capabilities of the proposed approach are demonstrated through representative boundary value problems.

Nonlinear 2D-DQ volume-preservative global–local dynamic analysis of composite sandwich plates with soft hyperelastic cores and viscoelastic Winkler-Pasternak foundations

Nonlinear large amplitude vibration and dynamic deformations of sandwich plates with composite face sheets and hyperelastic cores that encounter large thickness changes during the vibration are investigated here for the first time. Moreover, a new global–local zigzag hyperelastic plate theory that considers the incompressibility and large transverse deformability of the core, and employs transverse normal strains whose order is higher than that of the in-plane strains is proposed. The plate is resting on a viscoelastic Winkler-Pasternak substrate, exposed to various spatial and time distributions of loads, and experiencing a diversity of edge conditions. The governing equations of motion are obtained by using Hamilton’s principle, von Kármán’s assumptions, and the series expansion of the volume-preserving strain energy density function. A 2D differential quadrature (2D-DQ) technique is employed for the spatial discretization of a structure with both constitutive and kinematic nonlinearities, for the first time. The time-dependent combination of the nonlinear coupled motion equations and boundary conditions is treated by an iterative/updating Runge-Kutta time-marching technique and the effects of the thickness and aspect ratio of the plate, viscoelastic and shear parameters of the foundation, types of the edge conditions, constitutive parameters of the composite materials, fiber orientation angle, and magnitude and distribution pattern of the loads on the resulting dynamic lateral deflections are studied. Results reveal that while the vibration frequencies and amplitudes of the face sheets are quite different, the damping is more remarkable when stiffer cores are utilized. Moreover, the effects of the higher vibration modes are more apparent in looser edge conditions, smaller fiber angles, and special excitation frequencies.

Understanding the inelastic response of collagen fibrils: A viscoelastic-plastic constitutive model.

Collagen fibrils, which are the lowest level fibrillar unit of organization of collagen, are thus of primary interest towards understanding the mechanical behavior of load-bearing soft tissues. The deformation of collagen fibrils shows unique mechanical features; namely, their high energy dissipation is even superior compared to most engineering materials. Additionally, there are indications that cyclic loading can further improve the toughness of collagen fibrils. Recent experiments from Liu at al. (2018) focused on the response of type I collagen fibrils to uniaxial cyclic loading, revealing some interesting results regarding their rate-dependent and inelastic response. In this work, we aim to develop a model that allows interpreting the complex nonlinear and inelastic response of collagen fibrils under cyclic loading. We propose a constitutive model that accounts for viscoelastic deformations through a decoupled strain-energy density function (into an elastic and a viscous parts), and for plastic deformations through plastic evolution laws. The stress-stretch response results obtained using this constitutive law showed good agreement with experimental data over complex loading paths. Ultimately we use the model to gain more insights on how cyclic loading and rate effects control the interplay between viscoelastic and plastic deformation in collagen fibrils, and to extrapolate the results from experimental data, analyzing how complex cyclic load influences energy dissipation and deformation mechanisms. STATEMENT OF SIGNIFICANCE: In this work, we develop a viscoelastic-plastic constitutive model for collagen fibrils with the aim of analyzing the effects of inelasticity and energy dissipation in this material, and more specifically the competition between viscoelasticity and plasticity in the context of cyclic loading and overload. Experimental and theoretical approaches so far have not fully clarified the interplay between viscous and plastic deformations during cyclic loading of collagen fibrils. Here, we aim to interpret the complex nonlinear response of collagen fibrils and, ultimately, suggest predictive capabilities that can inform tissue-level response and injury. To validate our model, we compare our results against the stress-stretch data obtained from experiments of cyclic loaded single fibrils performed by Liu etal. (2018).

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Calculation of Strain Energy Density Function Using Ogden Model and Mooney-Rivlin Model Based on Biaxial Elongation Experiments of Silicone Rubber.

Strain energy density functions are used in CAE analysis of hyperelastic materials such as rubber and elastomers. This function can originally be obtained only by experiments using biaxial deformation, but the difficulty of such experiments has made it almost impossible to put the function to practical use. Furthermore, it has been unclear how to introduce the strain energy density function necessary for CAE analysis from the results of biaxial deformation experiments on rubber. In this study, parameters of the Ogden and Mooney-Rivlin approximations of the strain energy density function were derived from the results of biaxial deformation experiments on silicone rubber, and their validity was verified. These results showed that it is best to determine the coefficients of the approximate equations for the strain energy density function after 10 cycles of repeated elongation of rubber in an equal biaxial deformation state, followed by equal biaxial elongation, uniaxial constrained biaxial elongation, and uniaxial elongation to obtain these three stress-strain curves.

Numerical Simulation of Assembly Process and Sealing Reliability of T-Rubber Gasket Pipe Joints

Underground pipelines are vital parts to urban water supply, gas supply, and other lifeline systems, affecting the sustainable development of cities to a great extent. The pipeline joint, which is a weak link, may be seriously damaged during natural disasters such as earthquakes. The failure of pipe joints can cause leakage accidents, resulting in system failure and interruption, and even some secondary disasters. Herein, based on uniaxial and plane tensile test results of a T-rubber gasket material, the assembly process and sealing performance of a T-rubber gasket joint of a ductile iron pipe are numerically simulated using the Ogden third-order strain energy density function to fit the material constant. The simulation accounts for severe nonlinearities, including large deformations, hyperelasticity, and complex contacts. The effects of the assembly friction coefficient, assembly depth, and radial clearance deviation of the socket and spigot on the seal contact pressure are analyzed. The results suggest that the entire history of the deformation and stress variations during assembly can be clearly visualized and accurately calculated. For the different friction coefficients, the assembly depth corresponding to the sliding friction condition of the spigot pipe was 74 mm, while the minimum pushing force required to assemble the T-rubber gasket joint of a DN300 ductile iron pipe was 6.8 kN at the ideal situation with a friction coefficient of 0. The effective contact pressure of the rubber gasket seepage surface under various operating conditions is much higher than the normal pressure of municipal pipelines, thus indicating that the rubber gasket joint exhibits the ideal sealing performance. Furthermore, a certain deviation, which is about 20 mm, is allowed for the assembly depth of the rubber gasket joint such that the axial displacement of the pipe joint can be adapted under an earthquake or ground displacement.

Hyperelastic material modelling using symbolic regression

AbstractRecently, data‐driven approaches in the field of material modeling have gained significant attention. A major advantage of these approaches is the direct integration of experimental results into the models. Nevertheless, artificial neural networks (ANNs) are especially challenging to interpret from a physical point of view, since internal processes of ANNs are difficult to understand.In this work a new automatic method for the generation of constitutive models for hyperelastic materials is introduced. The presented method is based on symbolic regression, which is a genetic algorithm. Thereby, a mathematical model in the form of an algebraic expression is found that fits the given data as accurately as possible and has a compact representation. The strain energy density function is determined directly as a function of the strain invariants. The proposed ansatz is embedded into a continuum mechanical framework combining the benefits of known physical relations with the unbiased optimization approach of symbolic regression. Benchmark tests for the generalized Mooney‐Rivlin model for uniaxial, equibiaxial and pure shear tests are presented. Finally, the presented procedure is tested on a temperature‐dependent dataset of a thermoplastic polyester elastomer. A good agreement between obtained material models and experimental data is demonstrated.

Analytical-based exact-kernel vibration and long-term creep stress and large deformation redistributions of the suddenly pressurized incompressible visco-hyperelastic thick cylinders

The long-term creep and short-term damped dynamic redistributions of the large deformations and stresses are investigated here for long, suddenly pressurized, rubber-like visco-hyperelastic thick-walled cylinders, for the first time. The extension of the Mooney–Rivlin strain energy density function to incompressible visco-hyperelastic materials is accomplished by using a hereditary integral based on the Prony series and an exponential relaxation kernel. To extract a more accurate/efficient formulation, the relaxation kernel is processed instead of the common approximate discretization of the time derivative of the stress tensor in the convolution integral. The resulting nonlinear integrodifferential equations whose number of terms grows with time are solved by a novel analytical approach in conjunction with the second-order Runge–Kutta time-marching, bound-varying trapezoidal technique, and an iterative/updating scheme. In comparison to the commercial finite element analysis codes, the present analytical formulation and solution algorithm lead to higher accuracies from the mathematical point of view. The visco-hyperelastic properties are extracted from previously reported experimental results, using an error minimization fitting technique. Results reveal that in contrast to the hyperelastic vessel, the oscillations of the displacements and stresses of the visco-hyperelastic cylinder occur around levels that grow with time. Moreover, smaller relaxation parameters lead to smaller displacements and stresses and the responses intend earlier to the steady-state condition for higher values of the stiffness elements of the Prony series. In long term, the mean stresses and mean radial displacements of the inner and outer layers of the cylinder grow asymptotically with time but the largest overshoots and amplitudes of the responses occur at the early times. However, smaller values of the relaxation parameter expedite the convergence to the steady-state condition.

Fiber pull-out in a flexoelectric material

This work examines the fiber pull-out problem in the context of flexoelectricity. As a first approach we examine the deformation of an infinitely long prism due to the action of a concentrated out-of-plane line force. We assume that the material of the prism responds in the context of flexoelectricity and anti-plane conditions prevail. We incorporate the approach proposed by Giannakopoulos and Zisis [1] according to which the gradient of the polarization is incorporated into the strain energy density function. Accordingly, the field equations decouple in terms of the primary variables of the problem, that is the out-of-plane displacement and the polarization. The concentrated out-of-plane line force is materialized in the form of an out-of-plane body force. For the proposed problem we develop analytical solutions with respect to the out-of-plane displacement and polarization under static and dynamic conditions. Next, we propose a finite element (FE) model based on a structural plate analogue, for the solution of anti-plane flexoelectric boundary value problems. The proposed FE model is applied to the predefined problem and the numerical and the analytical results regarding the out-of-plane displacement are compared and found to be in excellent agreement. Next, we propose yet another structural analogue (membrane analogue) for the numerical treatment of the polarization equation and again the results are compared with the analytical solutions with respect to the polarization of the predefined problem and excellent agreement is attained. Finally, for the predefined problem the electromagnetic response is investigated and analytical solutions are proposed.

Flexible soft Pneumatic Bionic Hand Based on Multi-Jointed Structure

Compared with rigid robotic hands, soft hands can provide better safety and adaptability. In the process of gradual development, the multi-jointed structure that mimics the shape of the human hand has shown significant progress in realizing its personification. In this article, we propose a multi-jointed pneumatic soft hand, which is composed of multi-jointed soft fingers, thumb, thenar and 3D printed palm. It can express letters through sign language gestures and can grasp objects with different sizes, shapes, weights and surface textures. We tested the bending ability of the actuators under different air pressures to characterize the performance of actuators made of silicone rubber. Based on the strain energy density function of silicone rubber, the Yeoh model is used to calculate the relationship between the air pressure required and the bending angle. In addition, a dedicated pneumatic control system is designed and manufactured to enable the soft hand to automatically perform tasks set by the specific program. This new multi-jointed pneumatic soft hand has flexible bionic fingers, and has the advantages of fast response speed, low cost, easy manufacturing, assembly and replacement.

Modeling of Hyper-Viscoelastic Properties of High-Damping Rubber Materials during the Cyclic Tension and Compression Process in the Vertical Direction.

With the rapid development of the economy and urbanization, the construction of the urban rail transit system has had a great impact on the work, life, and health of residents in buildings along the rail transit line. Thus, it is particularly urgent and necessary to develop base isolation technologies to control and reduce the impact of vibrations of rail transit systems on building structures. High-damping rubber isolation bearings have shown significant effectiveness in the reduction of this impact, and their isolation performance mainly depends on the mechanical and damping energy dissipation characteristics of the high-damping rubber material. This paper aims to investigate the hyper-viscoelastic properties of the high-damping rubber material used for high-damping rubber isolation bearings during the cyclic tension and compression process in the vertical direction. These properties include hyperelastic parameters, viscoelastic coefficients, and the relaxation times of the material. For this purpose, uniaxial cyclic tension and compression tests were conducted. A three-element Maxwell rheological model combining a strain energy density function was proposed for modeling the hyper-viscoelastic behaviors of the materials during the cyclic tension and compression process. Based on the obtained results, an iterative identification procedure was used to determine the constitutive parameters of the material for each loading-unloading cycle. The aforementioned parameters were further expressed as a function of the number of cycles. New insights into hyper-viscoelastic property changes in this high-damping rubber material during the cyclic tension and compression process were gained in this work. These investigations could facilitate the development of computational tools, which would regulate fundamental guidelines for the better controlling and optimization of the isolation performance of the high-damping rubber material used for high-damping rubber isolation bearings, which have a wider perspective of applications in the urban rail transit system.

A viscoelastic-viscoplastic constitutive model for nanoparticle-reinforced epoxy composites: Particle, temperature, and strain rate effects

The mechanical behavior of nanoparticle-reinforced epoxy (NP/epoxy) exhibits highly nonlinear characteristics accompanied by particle, temperature, and strain rate effects, which are of great significance in the study of composites. This study presents a viscoelastic-viscoplastic (VE-VP) constitutive model applied to rigid silica and soft rubber nanoparticle reinforced epoxies (SNP/epoxy and RNP/epoxy), which are subjected to a wide strain rates and temperatures. The proposed constitutive model is based on the decomposition of the Helmholtz free energy into equilibrium and non-equilibrium parts. The equilibrium part employs the hyperelastic strain energy density function to capture the rubbery state property. The non-equilibrium part is decomposed into viscoelastic strain energy density functions and viscoplastic dissipation potential to capture the glassy state behavior. Temperature- and particle- dependent models were introduced into the proposed VE-VP model. To validate the effectiveness of the VE-VP model, the uniaxial compressive responses of the SNP/epoxy and RNP/epoxy were systematically tested. The results demonstrate that the VE-VP constitutive model can capture the nonlinear compressive behaviors of SNP/epoxy and RNP/epoxy over a strain-rate range of 10 − 2 s − 1 to 10 3 s − 1 and a temperature range of − 80 ℃ to 80 ℃ . Finally, the analysis of the model parameters indicates that the strain softening and hardening are strongly affected by the temperature and strain rate. • Two typical nanoparticles (rigid silica and soft rubber) are utilized to reinforced epoxy. • A viscoelastic-viscoplastic constitutive model for nanoparticles reinforced epoxy incorporating the particle, temperature and strain rate effects is proposed. • The compressive behaviors over wide range of temperature and strain rate are experimentally investigated and well predicted by proposed model. • The activation energy and hyperelastic coefficient in proposed model are strongly affected by temperature and strain rate.