Low Strain Rate Range Research Articles

Low strain rate mechanical performance of balsa wood and carbon fibre-epoxy-balsa sandwich structures

The focus of this study is the experimental assessment of the mechanical behaviour of balsa wood and its sandwich structures, where balsa serves as the core, supported by carbon fibre-epoxy skin layers. A comprehensive characterisation is conducted on the mechanical behaviour of balsa wood and carbon fibre-epoxy balsa core sandwich structures subjected to a range of low strain rates. Initially, the study undertakes a consistent procedure for sample preparation. Subsequently, the characterisation of the manufactured composite structures is performed experimentally. A stereo microscope is employed for a detailed visual inspection of the internal structure of the balsa wood and the sandwich structures. Furthermore, the mechanical characterisation is carried out with three-point bending tests at a range of strain rates from 0.1 % to 6 % strain per minute. This research reveals key findings about balsa wood and its sandwich structures, highlighting their performance and their sensitivity even under low strain rates.

Experiment and meso-scale modelling on combined effects of strain rate and specimen size on uniaxial-compressive failures of concrete

This paper presents experimental and meso-scale modelling studies on the combined effects of strain-rate and specimen size on uniaxial compressive failure behaviour of concrete. A series of uniaxial compressive tests and meso-scale simulations were conducted on concrete with various specimen sizes under different strain-rates covering the strain-rate range of seismic load, with special focus on the quantitative contribution of end-friction to compressive strength and its corresponding strain-rate effect as well as size effect. Results indicate that the uniaxial compressive failure with end-friction follows an hourglass failure pattern while that without end-friction exhibits a columnar failure pattern. The end-friction effect can form different confined zone distributions for various sized specimens, which can cause the contribution of end-friction to compressive strength is size dependent as well as enhance the influence of specimen size on static and dynamic strength. The contribution proportion of end-friction to compressive strength is around 20∼25%. Moreover, larger-sized specimen performs a stronger strain-rate effect and the increasing strain-rate can weaken the influence of specimen size on the real compressive strength. The proposed real DIF empirical formula considering the size-dependency (covering the low strain-rate range) can well estimate the strain-rate effect for concrete with different sizes, which can provide a valuable reference for the numerical calculation of dynamic mechanical response and the safety design of concrete structures.

Hot deformation behaviour of microstructural evolution in 2507 Super Duplex Stainless Steel during high-temperature straining

ABSTRACT In this study, the flow behaviour, and microstructure evolution of the SDSS2507 were systematically investigated via hot compressive tests. Subsequently, the hot processing map was constructed according to the obtained stress–strain curves and corresponding equations. Additionally, metallographic examinations of hot compressed specimens found that the softening mechanism of the ferrite and austenite is dynamic recovery (DRV) and dynamic recrystallisation (DRX), respectively. With increasing the strain rate, it promotes the DRX in austenite. Moreover, micro-cracks could initiate from the ferrite/austenite phase boundaries or grain boundaries within the ferrite due to the strong discrepancy of deformation coordination between the two phases or precipitation of Cr2N precipitates along the ferrite/austenite phase boundaries or ferrite grains at higher strain rate. Considering the hot processing map and microstructural evolution in SDSS2507, the optimum hot processing parameters for SDSS2507 are found to be in a low strain rate range of 0.1–3 s−1 with the temperature range of above 950°C.

Double-humped temperature dependence of plastic behaviour of K403 superalloy and its constitutive model

ABSTRACT In this study, uniaxial compression experiments of K403 superalloy were conducted over temperature range of 298–1073 K and strain rate range of 0.1–5200/s. Temperature and strain rate effects on the flow stress were analysed. Double anomalous stress humps were noticed in the flow stress vs. temperature curves. The flow stress of the superalloy showed independence on strain rate over low strain rate range and an enormous increase with increasing strain rate over high strain rate range. Then, the effects of temperature and strain rate on the microstructure were observed and analysed. Due to the double-humped temperature dependence, the existing constitutive models were unable to accurately predict the plastic behaviour of K403 superalloy. Thus, a constitutive model was developed to describe the plastic behaviour of K403 superalloy. The model was able to capture the double-humped temperature dependence and anomalous strain rate dependence of plastic behaviour. It was shown that the plastic behaviour of the superalloy can be accurately predicted over a wide range of temperatures and strain rates.

Single-Walled Carbon Nanotube-Enhanced Bagasse-Epoxy Hybrid Composites under Varied Low Tensile Strain Rates

The production demand of high-performance polymer composites utilizing natural and renewable resources, especially agricultural waste fibres, is rapidly growing. However, these polymers’ mechanical properties are strain rate-dependent due to their viscoelastic nature. Particularly, for natural fibre-reinforced polymer composites (NFPCs), the involvement of fillers has caused rather complex failure mechanisms under different strain rates. Moreover, unevenly and micro-sized bagasse-reinforced polymer composites often cause the formation of micro-cracks and voids in composites. Consequently, the rates of crack initiation and propagation of these composites become extremely sensitive. This, in turn, causes low and unpredictable tensile performance at higher tensile crosshead speeds, even within the low strain rate range. In this study, single-walled carbon nanotubes (SWCNTs) were applied to enhance the bagasse-epoxy composites’ strength. The effects of the weightage in the SWCNT loadings on the composites’ tensile properties were subsequently investigated under low strain rates of 0.0005 s−1, 0.005 s−1 and 0.05 s−1. The composites’ failure shifted to a higher distribution (65.7% improvement, from 37.23 to 61.68 MPa, across strain rates) due to the addition of 0.05% SWCNTs, as indicated in a Weibull distribution plot. The high aspect ratio and strong interface adhesion of SWCNTs in and toward the epoxy matrix contributed significantly to the composites’ strengths. However, a further increase in SWCNT content in the tested composites caused early embrittlement due to agglomeration. The toughness and characteristic strength improved significantly as the strain rate increased. A scanning electron microscopic (SEM) analysis revealed that the SWCNTs’ high aspect ratios and large surface areas improved the interface bonding between the filler and matrix. However, higher SWCNT loadings (0.15% and 0.25%) caused a reverse effect in the same properties of these composites under the same strain rate variations, due to agglomeration. Finally, an empirical relationship was developed to describe the strain rate effect of tensile properties containing 0.05% SWCNT-reinforced bagasse-epoxy composites.

Experimental Study on the Influence of Freezing Pressure on the Uniaxial Mechanical Properties of Ice

In this study, a test technique that enables continuous control of the sample stress state from freezing to testing is proposed to investigate the influence of freezing pressure on the mechanical properties of ice under uniaxial compression. In this method, the water is frozen into the standard cylindrical ice specimen under high hydraulic pressure in a triaxial pressure chamber, and then, the temperature field and stress field of the ice specimens are adjusted to the initial state of the test; finally, an in situ mechanical test is conducted in the triaxial chamber. The uniaxial compression test of ice specimens with temperature of −20°C and freezing pressure of 0.5–30 MPa is performed in the strain rate range of 5 × 10−5−1.5 × 10−6 s−1. The results show that, as the freezing pressure increases, the ductile-to-brittle transition zone of the ice specimen during failure moves to the low strain rate range, and the failure mode of the specimen changes from shear failure to splitting failure. Further, the brittleness index of the ice specimen first increases, then decreases, and then again increases with the increase in freezing pressure. The brittleness index reaches the maximum (minimum) when the freezing pressure is 30 MPa (20 MPa). The peak stress of the ice specimen also increases first, then decreases, and then increases with the increase in freezing pressure. The maximum value is also at the freezing pressure of 30 MPa, but the minimum value is obtained at the freezing pressure of 0.5 MPa. The failure strain of the ice specimen first decreases and then increases with the increase in freezing pressure, and the maximum (minimum) value is achieved at the freezing pressure of 0.5 MPa (10 MPa). When the ice specimen exhibits brittle failure, the relationships between the residual stress and the freezing pressure and between the peak stress and freezing pressure are the same, but when the ice specimen exhibits ductile failure, there is no obvious relationship between the residual stress and the freezing pressure.

Influences of Stress State, Temperature, and Strain Rate on Ductility of Pure Iron

In this study, the plastic response of pure iron DT8 at different stress states, temperatures, and strain rates is investigated. Specimens with different structures designed for different stress states are subjected to quasi-static uniaxial tension tests to investigate the effect of stress state on the fracture behavior of pure iron. Standard specimens at temperatures ranging from 100 to 900 °C are tested by a quasi-static uniaxial tension machine. Double-notch shear specimens are subjected to a universal electronic tensile testing machine and an improved split-Hopkinson pressure bar device with strain rates ranging from 0.0033 to 227450/s to investigate the effect of strain rate on the fracture behavior of pure iron. Results show that the flow stress of pure iron DT8 slightly increases with the increase in stress triaxiality before the material breaks. With the increase in stress triaxiality from 0.333 to 0.550, the fracture strain of pure iron DT8 decreases almost linearly from 1.949 to 0.898. With the increase in temperature, the fracture strain increases almost linearly from 1.43 at 100 °C to 3.19 at 900 °C. A blue brittleness effect occurs at about 300 °C, where the yield and tensile strengths of pure iron increase slightly. In the low strain rate range from 0.0033 to 800/s, the fracture strain of pure iron is almost stable at approximately 2.0. Then, the fracture strain increases with the further increase in strain rate. When the strain rate reaches 120850/s, the fracture strain reaches its peak value of 3.64. Thereafter, the fracture strain decreases rapidly as the strain rate increases. The fracture surfaces of the specimens take on different fracture morphologies with the increase in strain rate. The nature of the strain rate’s effect on material ductility is discussed.

A comparative study on hot deformation behaviours of low-carbon and medium-carbon vanadium microalloyed steels

Carbon is an essential element in steel, but there are still discrepancies regarding its effect on steel hot deformation behaviours. In this research, isothermal compression tests were carried out for a low-carbon (0.05 C) and a medium-carbon (0.38 C) vanadium microalloyed steel with deformation temperatures of 900−1050 °C and strain rates of 0.01−30 s−1. It was found that carbon causes a softening effect at low strain rates (0.01−1.0 s−1), while a hardening effect at high strain rates (10.0−30 s−1). Through constitutive analysis, the hot deformation activation energy for 0.05 C steel is 305.9 kJ/mol in the whole strain rate range, while for 0.38 C steel, the activation energy is 292.3 kJ/mol in the low strain rate range (0.01−1 s−1) and 475.0 kJ/mol in the high strain rate range (10−30 s−1). It was proposed that the addition of carbon decreases the deformation activation energy at low strain rates (0.01−1 s−1), due to its positive influence on the self-diffusion coefficient of iron, which increases the rates of dislocation climb and recovery. On the other hand, carbon lowers the stacking fault energy of austenite, which could make partial dislocation collapse more difficult than dislocation climb and thus becomes the rate-controlling mechanism for 0.38 C steel at high strain rates (10−30 s−1). This gives rise to the higher activation energy and work hardening rate of 0.38C steel at high strain rates. Comparing the power-dissipation-efficiency maps, two peak domains were found in those of 0.38 C steel, while only one peak domain exists in those of 0.05 C steel.

Direct Tensile Behavior of Limestone and Sandstone with Bedding Planes at Different Strain Rates

To investigate the tensile behavior of rock bedding planes, a series of direct tensile tests were conducted on cylindrical limestone and sandstone specimens with a distinct bedding plane perpendicular to the axial direction. Different tensile strain rates of the bedding plane areas within the magnitude range from 10−6 to 10−2 s−1 were considered. The results indicate that the limestone fracture surfaces maintain the original roughness of bedding planes with few grain fractures. However, the sandstone fracture surfaces display many intergranular and transgranular fractures and are rough at the grain scale. The stress–strain curves of limestone specimens can be divided into elastic deformation stage and yield stage (the nonlinearity of yield stage is weaker with the increase of strain rate), while that of sandstone specimens develop in a nonlinear way accompanied by plastic deformation throughout the loading process. The strain rate significantly affects the tensile deformation and strength of the tested specimens (except the average deformation modulus of sandstone specimens). The elastic modulus, average deformation modulus, peak strain and tensile strength increase as strain rate goes up and have a linear relation with the logarithm of strain rate under the tested strain rate range (namely, under a relatively low strain rate range).

The Low Strain Rate Response of As-Cast Ti-6Al-4V Alloy with an Initial Coarse Lamellar Structure

The microstructure and microtexture evolution of the as-cast Ti-6Al-4V alloy with an initial lamellar microstructure was investigated in the temperature range of 900–1050 °C and the low strain rate range of 10−3–10−1 s−1. In the ranges 900–950 °C and 10−3–10−2 s−1, globularization of the α lamellar structure took place; the initial α {0001} texture softened, and the {0001} texture rotated to the transverse direction. Additionally, the increasing strain rate led to more randomization. The globularization process was considered to be continuous dynamic recrystallization (CDRX). At strain rates higher than 10−2 s−1 and temperatures between 900 and 950 °C, flow instabilities occurred. In the β range, dynamic recrystallization (DRX) occurred at a temperature of 1050 °C and a strain rate of 10−3 s−1.

Mechanical Behavior of Two Ferrite–Martensite Dual-Phase Steels over a Broad Range of Strain Rates

The present study concerns the deformation and fracture behavior of two ferrite–martensite dual phase steels (FMDP660 and FMDP780) with different phase fractions subjected to different strain rate (0.001 s−1 to 1000 s−1) tensile testing. For both steels, the yield strength (YS) monotonically increased with strain rates, whereas the values of ultimate tensile strength (UTS), uniform elongation (UE) and post-uniform elongation (PUE) were maintained stable at the low strain rate range (0.001–0.1 s−1), followed by a significant increase with strain rate at high strain rate levels (0.1–1000 s−1). The FMDP780 steel with a higher fraction of martensite possessed a stronger strain rate sensitivity of tensile strength and elongation (UE and PUE) values at the high strain rate stage, compared with the FMDP660 sample. The change of UTS and UE with different strain rates and phase fractions was highly related to the strain hardening behavior, which was controlled by the dislocation multiplication in ferrite, as validated by transmission electron microscopy (TEM). The fracture surface of the two steels was characterized by dimpled-type fracture associated with microvoid formation at the ferrite–martensite interfaces, regardless of the strain rates. The change of the dimple size and PUE value of the two steels with strain rates was attributed to the effect of adiabatic heating during tensile testing.

Investigation on Mechanical Properties and Reaction Characteristics of Al‐PTFE Composites with Different Al Particle Size

Al‐PTFE (aluminum‐polytetrafluoroethylene) serves as one among the most promising reactive materials (RMs). In this work, six types of Al‐PTFE composites with different Al particle sizes (i.e., 50 nm, 1∼2 μm, 6∼7 μm, 12∼14 μm, 22∼24 μm, and 32∼34 μm) were prepared, and quasistatic compression and drop weight tests were conducted to characterize the mechanical properties and reaction characteristics of Al‐PTFE composites. The reaction phenomenon and stress‐strain curves were recorded by a high‐speed camera and universal testing machine. The microstructure of selected specimens was anatomized through adopting a scanning electron microscope (SEM) to correlate the mesoscale structural characteristics to their macroproperties. As the results indicated, in the case of quasistatic compression, the strength of the composites was decreased (the yield strength falling from 22.7 MPa to 13.6 MPa and the hardening modulus declining from 33.3 MPa to 25 MPa) with the increase of the Al particle size. The toughness rose firstly and subsequently decreased and peaked as 116.42 MJ/m3 at 6∼7 μm. The reaction phenomenon occurred only in composites with the Al particle size less than 10 μm. In drop weight tests, six types of specimens were overall reacted. As the Al particle size rose, the ignition energy of the composites enhanced and the composites turned out to be more insensitive to reaction. In a lower strain rate range (10−2·s−1∼102·s−1), Al‐PTFE specimens take on different mechanical properties and reaction characteristics in the case of different strain rates. The formation of circumferential open cracks is deemed as a prerequisite for Al‐PTFE specimens to go through a reaction.

Characterization of hot deformation behavior of Ti–3Al–5Mo–4.5V alloy with a martensitic starting microstructure

The hot deformation behavior of Ti–3Al–5Mo–4.5V alloy with an [Formula: see text] martensitic microstructure was studied in the temperature range of 700–1000[Formula: see text]C and strain rates range of 0.001–[Formula: see text] up to a height reduction of 50%. The results show that an ultrafine equiaxed microstructure with an average grain size of [Formula: see text] were successfully produced through thermomechanical processing of a martensitic starting microstructure. A processing map was successfully constructed and provides appropriate processing parameters for hot deformation which are located in the temperature range of 700–800[Formula: see text]C and the low strain rate range of 0.001–[Formula: see text]. The instability domain mainly occurs at the strain rate higher than [Formula: see text] for the whole deformation temperature, which should be avoided during practice. The flow softening mechanism of the alloy is determined to be continuous dynamic recrystallization and the unstable flow is caused by the macro-fracture and flow localization.

Increase in the plasticity of molybdenum after high-rate deformation under pressure

The dependence of the plasticity of polycrystalline molybdenum on the strain rate has been studied after deformation performed under pressure and at an ambient temperature of 293 K. The samples were deformed in tensile tests. The strain rate of tensile deformation and the pressure in each experiment are constant. The range of the strain rate is 8.3 × 10−7 to 8.3 m/s and the pressure is 0.1–500 MPa. At a pressure of ambient atmosphere, molybdenum at a strain rate less than 200 s–1 in value (102.3 s–1) fractures in a brittle manner with zero residual strain. The brittle fracture of the working part of a cylindrical sample occurs simultaneously in several places. The molybdenum plasticity decreases in the range of low strain rates regardless of the pressure and, on the contrary, plasticity under pressure increases in the range of high strain rates. The strain rate of tensile deformation at which the dependence of the plasticity on the strain rate under pressure changes is 8.3 × 10–4 m/s (a strain rate of 0.25 s–1). The high plasticity of molybdenum after deformation under pressure is observed at high strain rates.

유빙 및 평탄빙의 충돌에 의한 빙하중과 선체구조응답 해석기법

2 The most important factor in the structural design of ships and offshore structures operating in arctic region is ice load, which results from ice-structure interaction during the ice collision process. The mechanical properties of ice related to strength and failure, however, show very complicated aspect varying with temperature, volume fraction of brine, grain size, strain rate and etc. So it is nearly impossible to establish a perfect material model of ice satisfying all the mechanical characteristics completely. Therefore, in general, ice collision analysis was carried out by relatively simple material models considering only specific aspects of mechanical characteristics of ice and it would be the most significant cause of inevitable errors in the analysis. Especially, it is well-known that the most distinctive mechanical property of ice is high dependency on strain rate. Ice shows brittle attribute in higher strain rate while it becomes ductile in lower strain rate range. In this study, the simulation method of ice collision to ship hull using the nonlinear dynamic FE analysis was dealt with. To consider the strain rate effects of ice during ice-structural interaction, strain rate dependent constitutive model in which yield stress and hardening behaviors vary with strain rate was adopted. To reduce the huge amount of computing time, the modeling range of ice and ship structure were restricted to the confined region of interest. Under the various scenario of ice-ship hull collision, the structural behavior of hull panels and failure modes of ice were examined by nonlinear FE analysis technique.

The Role of Martensitic Transformation in Thermomechanical Development of Microstructure and Plasticity of Two-Phase Ti-6Al-4V Titanium Alloy

Phase constituent morphology in microstructure of two-phase α+β titanium alloys is determined by conditions of thermomechanical processing consisting of sequential heat treatment and plastic deformation operations. Results of previous research indicate that particularly solution treatment preceding plastic deformation significantly changes α-phase morphology and determines hot plasticity of titanium alloys. In the paper thermomechanical processing composed of β solution treatment and following hot forging of Ti-6Al-4V titanium alloy was analysed. Development of martensite plates during heating up and hot deformation was evaluated. Microscopic examinations revealed that elongated and deformed α-phase grains were fragmented and transformed into globular ones. Significant influence of martensitic transformation on elongation coefficient of α-phase grains after plastic deformation was confirmed. Based on results of elevated temperature tensile tests it was established that α-phase morphology in examined two-phase α+β titanium alloy, developed in the thermomechanical processing, can enhance their hot plasticity – especially in the range of low strain rates.

Hot Deformation Behavior and Processing Map of Ti-6Al-3Nb-2Zr-1Mo Titanium Alloy

Abstract The isothermal compression tests were performed over the ranges of temperatures 820∼1060 °C and strain rates 0.001∼1 s −1 on a Gleeble-3800 simulator. The constitutive equation of Ti-6Al-3Nb-2Zr-1Mo alloy has been established to describe the changing rule of flow stress with the strain rate and deformation temperature. The apparent activation energies have been calculated to be 764.71 kJ/mol in the dual phase region and 126.94 kJ/mol in the single phase region. The processing maps have been constructed based on the dynamic material model (DMM) and the Prasad's instability criterion at strains of 0.4 and 0.7. The maps exhibit a stable domain in the temperature range of 840∼1060 °C and strain rate range of 0.001∼0.1 s −1 with two peaks in power dissipation of 51%, occurring at 940 °C/0.001 s −1 and 880 °C/1 s −1 , respectively. The high efficiency values of power dissipation indicate dynamic recrystallization (DRX)/globalization in these fields. Based on the constructed processing maps, the optimal hot processing window of this alloy corresponds to the temperature range of 840∼1060 °C and the low strain rate range of 0.001∼0.1 s −1 . The microstructures of the specimens deformed at different conditions were analyzed and connected with the processing map. It is found that in processing of the alloy at 820 °C and a higher strain rate (≥1 s −1 ) an instability deformation will take place easily.

Modeling of the Strain Rate Dependency of Polycarbonate’s Yield Stress: Evaluation of Four Constitutive Equations

The main focus of this paper is in evaluating four constitutive relations which model the strain rate dependency of polymers yield stress. Namely, the two-term power-law, the Ree-Eyring, the cooperative, and the newly modified-Eyring equations are used to fit tensile and compression yield stresses of polycarbonate, which are obtained from the literature. The four equations give good agreement with the experimental data. Despite using only three material constants, the modified-Eyring equation, which considers a strain rate-dependent activation volume, gives slightly worse fit than the three other equations. The two-term power-law and the cooperative equation predict a progressive increase in the strain rate sensitivity of the yield stress. Oppositely, the Ree-Eyring and the modified-Eyring equations show a clear transition between the low and high strain rate ranges. Namely, they predict a linear dependency of the yield stress in terms of the strain rate at the low strain rate range. Crossing a threshold strain rate, the yield stress sensitivity sharply increases as the strain rate increases. Hence, two different behaviors were observed though the four equations fit well the experimental data. More experimental data, mainly at the intermediate strain rate range, are needed to conclude which, of the two behaviors, is more appropriate for polymers.

Influence of γʹ precipitates on Portevin–Le Chatelier effect of NI-based superalloys

The γ′ precipitate plays a critical role in improving the mechanical properties of Ni-based superalloys. An undesirable phenomenon referred to as the Portevin–Le Chatelier (PLC) effect always appears in Ni-based superalloys deformed within specific ranges of strain rate and temperature. In order to systematically investigate the influence of the γ′ precipitates on the PLC effect, four Ni-based superalloys with various γ′ contents were designed and fabricated. Microscopic observations from transmission electron microscopy (TEM) indicated that the volume fraction of the γ′ phase was consistent with the designed value. Furthermore, analysis of energy dispersive spectroscopy (EDS) results revealed that the γ matrix of all the alloys consisted of the same components. Uniaxial tensile tests were performed at strain rates and temperatures ranging from 1×10−4 to 3×10−3s−1 and 300–500°C, respectively. We found that the ultimate strength increased while the elongation decreased with increasing γ′ content. In addition, the serration changed from type A to type B and to type C with increasing temperature, decreasing strain rate or increasing γ′ content; the amplitude of type B serrations was described by unimodal or bimodal distributions. Increasing volume fraction of γ′ precipitates shifted the region in which the PLC effect occurred, to the range of low temperatures and high strain rates. Moreover, the serration amplitude increased with increasing γ′ content at a given temperature, which indicated that the γ′ precipitate increases the dynamic strain ageing (DSA) effect.

Strain rate dependent properties of ultra high performance fiber reinforced concrete (UHP-FRC) under tension

The results of an experimental investigation of UHP-FRC tensile response under a range of low strain rates are presented. The strain rate dependent tests are conducted on dogbone specimens using a hydraulic servo-controlled testing machine. The experimental variables are strain rate, which ranges from 0.00011/s to 0.11/s, fiber type, and fiber volume fraction. Five different types of fibers are considered including straight and twisted fibers with different geometric properties. The rate sensitivity of the composite material in tension is evaluated in terms of its first cracking strength, post-cracking strength, energy absorption capacity, strain capacity, elastic modulus, fiber tensile stress and number of cracks. The test results show pronounced rate effects on post-cracking strength and energy absorption capacity. Further, post cracking strength varies linearly with the fiber reinforcing index and energy absorption capacity varies linearly with the product of the fiber length and the reinforcing index, as predicted from the theory for fiber reinforced concrete.