Homogeneous Strain Field Research Articles

Temperature- and rate-dependent tensile behaviour of unidirectional carbon/polyamide-6 composite under off-axis loading with oblique tabs

Unidirectional carbon/polyamide-6 (CPA6) is a thermoplastic composite with attractive properties for many industrial applications. However, shortage of experimental data may hinder the development of reliable material models. This paper is the first to provide a complete dataset for the tensile behaviour of unidirectional CPA6, including the longitudinal, transverse, in-plane shear, and off-axis response, at different temperatures and (quasi-static) strain rates. The oblique end tab design, already proven in the literature for off-axis testing of composites at room temperature, was successfully extended to the elevated temperature of 120 °C. The use of cardboard tabs and fast-curing adhesive greatly simplified the tab manufacturing and application for all the combinations of fibre angles, temperatures and strain rates. Digital image correlation (DIC) was performed through the window of the temperature chamber to verify the homogeneous strain field produced by the oblique tabs. The in-plane shear behaviour was extracted from the off-axis tests, allowing to estimate the shear modulus and shear strength. Finally, the robustness of the experimental results generated in this study was demonstrated with well-established analytical models that could excellently predict the elastic modulus, Poisson's ratio, and failure stress at different fibre angles.

A multiscale constitutive model of magnesium-shape memory alloy composite

In this work, a multiscale constitutive model is established to describe the deformation behaviors of magnesium-shape memory alloy (Mg-SMA) composite in a wide temperature range and reveal the strengthening mechanism of SMA reinforcement on Mg. The model is established at the grain scale firstly and gradually transited to the macroscopic scale by employing a newly developed three-level scale transition rule. At the grain scale, the thermodynamic-consistent constitutive models of Mg and SMA are, respectively, constructed by addressing different inelastic deformation mechanisms. The basal, prismatic, pyramidal, slip systems and extension twinning system are considered for the Mg phase, and the martensite transformation (MT) and austenitic plasticity are addressed for SMA reinforcement. Thermodynamic driving forces of each inelastic deformation mechanism are derived from the dissipative inequality and the constructed Gibbs free energies. At the polycrystalline scale, to evaluate the interactions among the grains and pores, and obtain the whole responses of the polycrystalline Mg and SMA, a thermo-mechanically coupled self-consistent homogenization scheme is employed. At the mesoscopic scale, a modified thermo-mechanically coupled Mori-Tanaka's homogenization scheme is adopted to evaluate the interaction between the Mg phase and SMA phase, and predict the whole responses for the representative volume element (RVE) of the composite. According to the geometrical features and mechanical loadings applied on the specimen, a hypothesis of homogeneous stress and strain fields at the macroscopic scale is adopted to achieve the scale transition from the RVE of the composite to the whole specimen. The capacity of the multiscale model is verified by comparing the predictions with the existing experimental data (Aydogmus, 2015). Moreover, the influences of characteristic information for the microstructures at different spatial scales on the deformation behaviors of the composite are predicted and discussed.

Viewing SiGe Heterostructure for Qubits with Dislocation Theory

SiGe/Si/SiGe heterostructure serves as one of the ideal semiconductor material system to build large scale electron spin qubits[1]. To further push the scalability and extend the coherence times of qubits built on SiGe/Si/SiGe heterostructures, there are several factors to improve, one of that is the lattice homogeneity of strained Si. One source of lattice homogeneity break down are dislocations. Due to the difficulties of high quality SiGe bulk crystal growth [2-3], relaxed SiGe is commonly grown on Si substrates by epitaxy. Viewing the SiGe/Si/SiGe heterostructure on native Si substrate, dislocations half loops form unavoidably because of the misfit between SiGe and the Si substrate. These dislocation half loops consist of threading segments extending through the whole heterostructure (figure 1 (a) and (c)) [4]. Due to the stress caused by the strained Si quantum well layer, the threading dislocations can be driven to glide, resulting in additional misfit segments in the lower interface of strained Si layer when the critical thickness is exceeded (figure 1 (b)) [5]. The extension of these misfit segments is blocked when they meet other perpendicular dislocations (figure 1 (d)) [6]. These misfit dislocations destroy strain uniformity and cause tilt inside the Si layer, resulting in a distortion of the valley splitting and ultimately, reducing the coherence time of qubits [7-8].In this work, we present the observation of dislocations in SiGe/Si/SiGe heterostructures for qubit application by plan-view TEM (Figure 1 (right)). Here, the experimental observations agree with dislocation theory that threading dislocations (some are marked by (b)) will glide and elongate to have misfit segments (some are marked by (c)) in the interface when two requirements are fulfilled: misfit exists between epitaxial layer and substrate; critical thickness is exceeded. When the misfit dislocations meet perpendicular misfit dislocation line, they are blocked since the thickness is not enough to glide over it [6]. This is the case in figure 1 right marked by (d). Due to the misfit segment generation and blocking effects, the misfit dislocation spacing is proportional to the threading dislocation density both theoretically and experimentally. These misfit dislocations bring local strain field around them [9], which disturb the desired homogeneous strain field by misfit. Therefore, misfit dislocation generation needs to be suppressed if one wants to achieve homogeneous strain in Si thin layer in large scale.According to Matthews-Blakeslee theory, one can avoid the formation of misfit dislocations in the Si interface by controlling the strained layer within the critical thickness. The critical thickness of Si on Si0.7Ge0.3 is calculated as 8.5 nm. The Electron channeling contrast images of Si with difference thickness 10 nm and 5 nm grown on high threading dislocation relaxed Si0.7Ge0.3 buffer at 700°C by reduced pressure chemical vapor deposition in Figure 2 are collaborating with theory: obviously misfit dislocation appear in 10 nm Si on relaxed Si0.7Ge0.3 (Figure 2 (left)) and no misfit dislocation are presented in 5 nm Si on Si0.7Ge0.3 (Figure 2 (right)).Our results show that dislocation generation and movement needs to be taken into consideration to further develop SiGe heterostructure based qubits.

Influence of accelerated corrosion on bi-material steel-CFRP double-lap joints bonded with thick adhesive

Bi-material steel-composite joints attract interest for marine applications. The marine environment imposes corrosion which is a prolonged process. This work presents a two-electrode electrochemical setup for accelerating free corrosion of the steel surfaces of bi-material joints. It is used to study the impact of accelerated corrosion on the mechanical performance of bi-material double lap specimens subjected to quasi-static tensile testing. Several test series have been conducted to evaluate the influence of overlap length and bond line thickness on shear strength and failure mode. Sixty specimens with a thick layer of methyl methacrylate adhesive have been fabricated and cured at room temperature for at least 24 h. Subsequently, 30 specimens were aged by subjecting them to electrochemical corrosion for 24 h. All specimens were tested for failure in quasi-static tensile loading while monitoring the strain fields in the joint area using digital image correlation. The measurements reveal a homogeneous shear strain field at the onset of loading, with a rapidly increasing shear strain concentration near the edges of the bond line preceding final failure. Both a decrease in the adhesive thickness and an increase in the overlap length increase the shear strength. Higher shear strength was observed for the electrochemically aged specimens than that of non-aged specimens. This is attributed to the faster residual curing of the adhesive during ageing because an increasing percentage of copper ions (released from the anode) accelerates the curing of the MMA adhesive. The electrochemically aged specimens showed mixed failure modes, i.e. cohesive failure and adhesive failure at the interface between steel and adhesive, and skin failure within the composite laminates.

Investigating the plastic anisotropy and hardening behavior of a commercial Zn–Cu–Ti alloy: Experimental & modeling approach

The aim of this work is to conduct a comprehensive examination of the anisotropic plastic behavior of a sheet of commercial Zn–Cu–Ti alloy, undergoing different quasi-static deformation paths at room temperature. Various mechanical tests are performed, including uniaxial and bi-axial tensile tests, Meuwissen and shear tests, obtaining homogeneous and heterogeneous strain fields. It was observed that under monotonic loading, the material behavior, i.e. stress evolution and ductility, is very sensitive to the loading direction. During reverse loading, the material displays a high Bauschinger effect, with behavior that is strongly dependent on the forward shear direction. For the Meuwissen-like tests, it has been found that the force–displacement curve as well as the maximum ductility are very sensitive to the in-plane anisotropy. The yield parameters are identified by direct comparison of the experimental results with classical anisotropic yield models. Finally, the plastic anisotropic behavior of the investigated zinc sheet is simulated for various mechanical tests using Finite Element Analysis to identify the related hardening parameters.

Multiscale Modelling and Mechanical Anisotropy of Periodic Cellular Solids with Rigid-Jointed Truss-Like Microscopic Architecture

This paper investigates the macroscopic anisotropic behavior of periodic cellular solids with rigid-jointed microscopic truss-like architecture. A theoretical matrix-based procedure is presented to calculate the homogenized stiffness and strength properties of the material which is validated experimentally. The procedure consists of four main steps, namely, (i) using classical structural analysis to determine the stiffness properties of a lattice unit cell, (ii) employing the Bloch’s theorem to generate the irreducible representation of the infinite lattice, (iii) resorting to the Cauchy–Born Hypothesis to express the microscopic nodal forces and deformations in terms of a homogeneous macroscopic strain field applied to the lattice, and (iv) employing the Hill–Mandel homogenization principle to obtain the macro-stiffness properties of the lattice topologies. The presented model is used to investigate the anisotropic mechanical behavior of 13 2D periodic cellular solids. The results are documented in three set of charts that show (i) the change of the Young and Shear moduli of the material with respect to their relative density; (ii) the contribution of the bending stiffness of microscopic cell elements to the homogenized macroscopic stiffness of the material; and (iii) polar diagrams of the change of the elastic moduli of the cellular solid in response to direction of macroscopic loading. The three set of charts can be used for design purposes in assemblies involving the honeycomb structures as it may help in selecting the best lattice topology for a given functional stiffness and strength requirement. The theoretical model was experimentally validated by means of tensile tests performed in additively manufactured Lattice Material (LM) specimens, achieving good agreement between the results. It was observed that the model of rigid-joined LM (RJLM) predicts the homogenized mechanical properties of the LM with higher accuracy compared to those predicted by pin-jointed models.

Three-dimensional strain descriptors at the Earth’s surface through 3D retrieved co-event displacement fields of differential interferometric synthetic aperture radar

In deformation analysis, invariant strain descriptors that are independent of the coordinate system are significant. An approach is proposed to determine the strain tensor and invariant strain quantities through the 3D retrieved co-event displacement of differential interferometric synthetic aperture radar (DInSAR). An optimal cell count is selected based on the proposed rule-of-thumb formula. This set of coherent cells with prescribed acceptable minimum value coherence is distributed on the domain through a quasi-random number generator. The recursive moving least squares method is applied to estimate the components of strain and differential rotation tensors. In practice, this method is not limited to the homogeneous strain field hypothesis. The objective here is to propose an approach to determine the local and regional strain patterns in SAR’s image resolution details. This approach is tested on both synthetic and real data. The real data consist of 3D displacements of 19 global navigation satellite system stations and SAR images of Kilauea Volcano eruption in July 2007. Results indicate that the areas with maximum shear strain and maximum dilatation are located exactly near the east rift zone of the volcano.

Deformation inhomogeneity at the crack tip of polycrystalline copper

Molecular dynamics simulation is conducted on single crystal and polycrystalline copper. Two different crack tip orientations (001) [-110] and (001) [010] are selected for copper single crystal. For copper single crystal with crack tip orientation of (001) [010], stress is concentrated at the crack tip, and cleavage crack propagation happens due to lack of dislocation activity at the crack tip. For copper single crystal with crack tip orientation of (001) [-110], continuous emission of dislocations from crack tip results crack tip blunting and reduction of stress concentration as well. Homogeneous crack tip stress and strain fields are evident for copper single crystals, however, stress and strain fields slightly alter with the generation of dislocations and defects. Crack tip stress and strain distribution are inhomogeneous from the beginning of the deformation for polycrystalline copper. Higher stress concentration at the grain boundary than crack tip results in nucleation of initial dislocations from the grain boundary during plastic deformation.

Cruciform Specimen Machining Using EDM and a New Design Verification for Biaxial Testing

Cruciform flat specimen is mostly used to investigate experimentally the in-plane biaxial mechanical behavior of sheet metals. In this study, a modified cruciform specimen design is proposed and validated against the performance of current standard design. The existing standard cruciform specimen prepared through laser cutting for sheet metal can accurately provide biaxial stress–strain curve through biaxial tensile test by ensuring superior homogeneous stress and strain field in gauge section. However, preparation of such specimen using laser cut is cumbersome, expensive and prone to errors. The current standard cruciform specimen design features slotted arm of fixed length and width with specific gauge. Few other designs also feature reduced gauge section to localize failure. Both need high precision of laser cutting and machining to prepare specimens which can achieve biaxial stress–strain curve for more than 4% equivalent plastic strain level. The proposed specimen design features running slits in all four arms of specimen with uniform gauge section machined using electrical discharge machining (EDM) owing to higher accuracy and minimum heat-affected zone (HAZ). This technique also avoids machining effects with an understanding that proposed specimen design is to determine the yield behavior of material only. Also, specimen fabricated from as received sheets and without any thickness reduction in gauge area allows the user to keep full thickness in biaxial condition. The modified design qualifies the commonly accepted performance criteria for cruciform specimen, i.e., stress and strain distribution in the gauge section is found to be homogeneous and the biaxial stress–strain curve can be generated until 4% of equivalent plastic strain. The proposed specimen design is evaluated experimentally by conducting biaxial test at various stress ratios, and it is compared with alike experiments using standard specimen design. One automotive grade material, interstitial-Free high-strength (IFHS) steel sheet of thickness 0.70 mm is tested in this study to validate the specimen design.

Experimental investigation of elastic wave propagation and inertia effects in extra soft gelatin material

In this paper, elastic wave propagation and inertia effects in gelatin-based soft material under impact loading is investigated. The dynamic test is principally performed by the classical SHPB (Split Hopkinson Pressure Bars) technique with highly sensitive piezoelectric polyvinylidene fluoride (PVDF) pressure sensors placed on the interfaces between the specimen and the bars. Digital image correlation (DIC) technique is conducted to measure the strong non- homogeneous strain field, the particle velocity, the the strain rate, the strain acceleration under transient stress state using high speed photography. The particle velocity and its gradient are used to extract the elastic wave velocity. Additionally, we derive closed-form equations for the additional axial stress produced by transverse inertia in the rectangular parallelepiped specimen under the non-equilibrium stress state. The amplitude of transverse inertial stress depends strongly on the strain rate and the strain acceleration. The transverse inertia results in a significant spike-like stress in the initial portion of inertial stress history and in the zone of the specimen near the impact side due to the acceleration stage of the impact test. It is obvious that the degree of inertia effects depends strongly on time steps and spatial positions.

Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.

Comparison of inverse identification strategies for constitutive mechanical models using full-field measurements

The calibration of phenomenological constitutive material models has been a constant need, because the parameters differ for each material and the ability of a model to mimic the real behaviour of a material is highly dependent on the quality of these parameters. Classically, the parameters of constitutive models are determined by standard tests under the assumption of homogeneous strain and stress fields in the zone of interest. However, in the last decade, Digital Image Correlation techniques and full-field measurements have enabled the development of new parameter identification strategies, such as the Finite Element Model Updating, the Constitutive Equation Gap Method, the Equilibrium Gap Method and the Virtual Fields Method. Although these new strategies have proven to be effective for linear and non-linear models, the implementation procedure for some of them is still a laborious task. The aim of this work is to give a detailed insight into the implementation aspects and validation of these methods. Detailed flowcharts of each strategy, focusing on the implementation aspects, are presented and their advantages and disadvantages are discussed. Moreover, these modern strategies are compared for the cases of homogeneous isotropic linear elasticity and isotropic plasticity with isotropic hardening. A simple numerical example is used to validate and compare the different strategies.

Experimentally Achievable Accuracy Using a Digital Image Correlation Technique in measuring Small-Magnitude (

Measuring small-magnitude strain fields using a digital image correlation (DIC) technique is challenging, due to the noise-signal ratio in strain maps. Here, we determined the level of accuracy achievable in measuring small-magnitude (<0.1%) homogeneous strain fields. We investigated different sets of parameters for image processing and imaging pre-selection, based on single-image noise level. The trueness of DIC was assessed by comparison of Young’s modulus (E) and Poisson’s ratio (ν) with values obtained from strain gauge measurements. Repeatability was improved, on average, by 20–25% with experimentally-determined optimal parameters and image pre-selection. Despite this, the intra- and inter-specimen repeatability of strain gauge measurements was 5 and 2.5 times better than DIC, respectively. Moreover, although trueness was also improved, on average, by 30–45%, DIC consistently overestimated the two material parameters by 1.8% and 3.2% for E and ν, respectively. DIC is a suitable option to measure small-magnitude homogeneous strain fields, bearing in mind the limitations in achievable accuracy.

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.

A new specimen setup for high precision uniaxial tension-compression tests of rubber

The aim of this contribution is the development of a new test specimen, which allows the investigation of rubber in combined uniaxial tension-compression tests. In comparison to the standard dumbbell, which also allows combined tests, the homogeneity in the measuring zone is significantly improved. Furthermore, the achievable amount of compression is increased substantially. The test specimen itself has a slender dumbbell shape. By a special design of the mounting geometry, a homogeneous strain field can be achieved. With the developed specimen setup, the phenomenological characteristics of rubber like Payne-effect, Mullins-effect, recovery and relaxation behavior can be investigated in more detail.

Effective X-ray elastic constant of cast iron

X-ray diffraction is a non-destructive method used for strain measurements in crystalline materials. Conversion of strain to stress can be achieved using the X-ray elastic constants (XEC), s1 and ½s2. The sin2ψ method was used during in situ loading to determine XEC for flake, vermicular, and spherical graphite iron. A fully pearlitic steel was used as reference. Uniaxial testing was conducted on the cast iron to create a homogeneous strain field, as well as four-point bending in both tension and compression due to the tension/compression asymmetry. The commonly used XEC value ½s2 = 5.81 × 10−6 MPa−1 is theoretically derived from an α-Fe single crystal. When investigating materials that contain ferrite, such as polycrystalline cast iron, this value is not accurate. Determination of an effective XEC for polycrystalline cast iron yields a better correlation between the measured microstrains and the properties observed on a macroscopic scale. The need for an effective XEC is evident, especially when it comes to model validation of, for example, casting simulations. Effective XEC values have been determined for flake, vermicular, and spherical graphite iron. The determined value is lower than the theoretical value.

General method for simulating laboratory tests with constitutive models for geomechanics

SummaryThe paper describes an approach to the simulation of arbitrary laboratory tests with homogeneous stress and strain fields applicable to most constitutive models common in geomechanics. The method by Bardet and Choucair [Int. J. Numer. Anal. Meth. Geomech. 15(1):1–19, 1991] is generalized for an arbitrary number of controlling and controlled variables. The approach is illustrated for time‐dependent thermo–hydro–mechanical constitutive models. The purpose of the method is to define an endless variety of different laboratory tests declaratively by means of two matrices E and S, which define the constraints on the controlled quantities such as stress, strains, suction, or temperature. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Homogeneous shear stress field of wood in an Arcan shear test configuration measured by means of electronic speckle pattern interferometry: description of the test setup

Several studies have dealt with the problem of how to measure the shear modulus of small clear wood specimens, avoiding bias from normal compression, bending, tension or torsional stresses. Unbiased results can be used to estimate realistic shear modulus values relevant for timber construction. However, a stress field that contains only uniformly distributed shear stress cannot be achieved. The Arcan shear test is one of the test setups that allows the measurement of more or less homogeneous shear stress and, therefore, shear deformation in a small volume of the specimen at macroscopic material level. The shear deformation of Arcan shear test samples was measured between the two notches by means of electronic speckle pattern interferometry (ESPI). Shear tests were performed in a tension mode, which provided a homogeneous strain field in the reduced midsection. Twisting forces and inhomogeneous shear deformation of the sample volume were assumed, which resulted in different shear deformation on the front and back sides of the sample. In this way, both surfaces of the samples were measured in parallel. Using the mean value of both deformation fields allowed a significant reduction in the coefficient of variation of shear modulus measurements to be achieved in comparison with values gained from a single ESPI measurement.

Impact of homogeneous strain on uranium vacancy diffusion in uranium dioxide

We present a detailed mechanism of, and the effect of homogeneous strains on, the migration of uranium vacancies in ${\mathrm{UO}}_{2}$. Vacancy migration pathways and barriers are identified using density functional theory and the effect of uniform strain fields are accounted for using the dipole tensor approach. We report complex migration pathways and noncubic symmetry associated with the uranium vacancy in ${\mathrm{UO}}_{2}$ and show that these complexities need to be carefully accounted for to predict the correct diffusion behavior of uranium vacancies. We show that under homogeneous strain fields, only the dipole tensor of the saddle with respect to the minimum is required to correctly predict the change in the energy barrier between the strained and the unstrained case. Diffusivities are computed using kinetic Monte Carlo simulations for both neutral and fully charged state of uranium single and divacancies. We calculate the effect of strain on migration barriers in the temperature range 800\char21{}1800 K for both vacancy types. Homogeneous strains as small as $2%$ have a considerable effect on diffusivity of both single and divacancies of uranium, with the effect of strain being more pronounced for single vacancies than divacancies. In contrast, the response of a given defect to strain is less sensitive to changes in the charge state of the defect. Further, strain leads to anisotropies in the mobility of the vacancy and the degree of anisotropy is very sensitive to the nature of the applied strain field for strain of equal magnitude. Our results suggest that the influence of strain on vacancy diffusivity will be significantly greater when single vacancies dominate the defect structure, such as sintering, while the effects will be much less substantial under irradiation conditions where divacancies dominate.

Determination of Precise Optimal Cyclic Strain for Tenogenic Differentiation of Mesenchymal Stem Cells Using a Non-uniform Deformation Field

Although there have been a number of studies regarding tenogenic differentiation of human bone marrow mesenchymal stem cells (hBMSCs) using uniaxial cyclic stretching stimulation with a homogeneous strain field, it has been difficult to determine the optimal normal strain in the stretch direction for differentiation. We have developed a non-uniform strain field system that in principle would allow determination of the optimal normal strain in a single experiment. As a result, the optimal normal strains of the membrane were clarified as 7.9 and 8.5 % for tenogenic differentiation marker proteins, type I collagen (Col I) and tenascin-C (Tnc), respectively. The non-uniform strain field proposed here represents a powerful tool to determine the optimal normal strain in the stretch direction for hBMSC-to-tenocyte differentiation. Furthermore, we found that the dependence of protein expression level on the normal strain of the membrane differed between proteins, which would be crucial in the field of embryology and regenerative medicine.