A novel formulation of material nonlinear analysis in structural mechanics

Himanshu Gaur .Anupam Srivastav

a Institute of Structural Mechanics.Bauhaus-Universit¨at Weimar.Marienstrasse 15.D-99423.Weimar.Germany

b Department of Civil Engineering.Middle East College.Muscat.Oman

Keywords: Computational methods Material nonlinear analysis Fracture mechanics Energy release rate etc

ABSTRACT This article demonstrates a novel approach for material nonlinear analysis.This analysis procedure eliminates tedious and lengthy step by step incremental and then iterative procedure adopted classically and gives direct results in the linear as well as in nonlinear range of the material behavior.Use of elastic moduli is eliminated.Instead.stress and strain functions are used as the material input in the analysis procedure.These stress and strain functions are directly derived from the stress-strain behavior of the material by the method of curve fitting.This way.the whole stress-strain diagram is utilized in the analysis which naturally exposes the response of structure when loading is in nonlinear range of the material behavior.It is found that it is an excellent computational procedure adopted so far for material nonlinear analysis which gives very accurate results.easy to adopt and simple in calculations.The method eliminates all types of linearity assumptions in basic derivations of equations and hence,eliminates all types of possibility of errors in the analysis procedure as well.As it is required to know stress distribution in the structural body by proper modelling and structural idealization.the proposed analysis approach can be regarded as stress-based analysis procedure.Basic problems such as uniaxial problem.beam bending.and torsion problems are solved.It is found that approach is very suitable for solving the problems of fracture mechanics.Energy release rate for plate with center crack and double cantilever beam specimen is also evaluated.The approach solves the fracture problem with relative ease in strength of material style calculations.For all problems.results are compared with the classical displacement-based liner theory.

1.Introduction

Nonlinear finite element analysis is derived a long back[1,2].At that time.all the derivations and calculations were known for the linear analysis only.An obvious approach that was adopted for solving a nonlinear analysis problem was derived from these equations and derivations of this linear analysis.The result of this approach is today′s lengthy and tedious step by step and then iterative procedure[3-8].

In this article.a novel approach is presented for solving a structural problem in the nonlinear range of the material behavior.If the stress distribution within the structural element is defined with proper structural idealization and modelling assumptions,then this methodology can be suitably applied.Therefore.the approach can be regarded as stress-based structural analysis approach.It is indifferent to the displacement-based analysis approach adopted classically.The key to this approach is to avoid linear mapping of deformation with geometry of the structural body which is adopted so far for the analysis procedures of the structural problems [9,10].In order to accommodate the effects of large deformation.true stress and true strain are used instead of engineering stress and engineering strain.

The proposed methodology is able to give ideal results either it is a linear or nonlinear range of material behaviour.This methodology eliminates the use of elastic moduli which can be considered as the approach for linear analysis only.Methodology utilizes the whole stress-strain behavior of the material in terms of stress and strain functions which are directly derived from the experimentalstress-strain data of the material by the method of curve fitting.These stress and strain functions work very much like the energy conjugates such as Green-Lagrange strain and Piola-Kirchhoff stress used in the conventional nonlinear finite element analysis[11].

It is found that the methodology is very simple to adopt,and it eliminates all the linearity assumptions in the derivations of basic equations.The proposed stressed based analysis procedure is perhaps challenging the whole development of nonlinear analysis procedure developed so far because of its ease.simplicity.and accuracy of results.It is a clear in fundamentals and gives clear physical picture/basics of the nonlinear analysis procedure.

2.Fundamental approaches of structural analysis

Basics of structural analysis start with simple static equations of equilibrium of a structural element.These static equations lead to differential equations of equilibrium together with boundary conditions.These differential equations are force equilibrium i.e.internal stresses generated in the structural member balances external loads.In order to solve the structural problem.material property elastic modulus is used to resolve the problem for the desired solution [12-15].

In today′s practice.finite element analysis is a strong formulation that is used abundantly for solving a class of problems such as structural mechanics.fluid mechanics.thermodynamics and electromagnetic,etc.almost all the phases of science and engineering.Finite element formulation is the simple extension of Weighted Residual methods or Ritz method.These methods give an approximate solution to the structural problem at hand [16-24].

Finite element formulation and its preliminary development i.e.weighted residual method or Ritz methods etc.together with some basic theories proposed in the literature.for example.Euler- Bernoulli Beam theory or Timoshenko Theory of beam bending.etc.make some kinematic assumptions prior to the solution of the problem.These kinematic assumptions are fundamentally assumed/proposed solutions of the structural problem at hand.For example.shape functions in isoparametric finite element formulation and assumed solution satisfying boundary conditions in Ritz method [25-27].The key point to notice here is that these all methods used so far,used elastic moduli as the material property/input for solving the structural problem.Elastic moduli are just the ratio of stress versus strain and the solution found by these approaches is applicable till the linear range of the material behaviour.If one wants to solve the problem further.it leads to step by step and then iterative procedure which is lengthy and tedious and uses the secant modulus [1].

Finding the solution by the theory of elasticity style also uses equations of equilibrium.It considers material as continua and utilizes compatibility conditions with boundary conditions and solves the structural problem mathematically which satisfies the equations with mathematical expertise.Examples are solution by Airys Stress functions and Westergaard′s solution for cracks etc.[28,29].Finding the solution with this approach gives a very accurate and precise solution.This approach also utilizes elastic moduli as the material input for solving the structural problem and hence fails in finding the solution in nonlinear range of the material behavior.This approach is very precise and lengthy.and perhaps approximate solutions should be sufficient for engineering and design purposes.

In the discussion so far.it is to note that these all fundamental approaches of structural mechanics use force balance i.e.external load applied counter balances internal stresses generated in the structural element which is.in fact.the differential equations of equilibrium of the problem at hand.In the proposed approach,we utilize energy balance approach.Use of elastic moduli it prohibited and the whole stress-strain curve is utilized in the calculations.

3.Applications in fracture mechanics

Early developments in fracture mechanics were started after Griffith theory when a new parameter was introduced as ‘energy release rate’ [30,31]Those early developments in fracture mechanics were essentially made for the highly brittle materials such as glass.This analysis procedure is not very suitable and effective when it comes to elastic materials such as metals.which possess high amount of plasticity.Unfortunately.during the course.this viewpoint was recognized later as limiting and unnecessary[32,33].

Another complicacy in the current state of art of fracture mechanics is the only available approach for nonlinear analysis i.e.step by step and then iterative procedure.Examples are Lagrangian and Total Lagrangian formulation [1].It is evident that as the material fractures,it definitely undergoes plastic deformation if it possesses the plasticity.It is because of these classical approaches for nonlinear analysis,Compliance approach is used in deriving energy release rate for double cantilever beam specimen.Even in the basic derivation of J- Integral.total potential of the system is derived either by considering (i) Load remains constant and displacement increases as crack grows or(ii)Displacement remains constant and load increases as the crack grows.which is not very acceptable in real physical sense [33-35].

The parameter Energy Release Rate(G)in fracture mechanics is utilized to determine the energy required to break the intermolecular bond between the molecules of the material.in order to propagate a crack surface.This energy based approach of crack propagation is considered to be the birth of fracture mechanics[30,31,36,37].Interestingly,this energy release rate can be suitably calculated by the principles of structural mechanics [38,39].

Energy based methods are abundantly used in fracture mechanics for simulating crack and its propagation till date [40-44].One of the pioneering work in this direction is done by Hillerborg et al..in 1976 when the crack was modelled with finite element analysis for concrete material.The model was based on energy balance approach [45].Later.in 2005 the work was extended further by Timon Rabczuk et al.when the concrete in compression was described by a non-local isotropic damage constitutive law and for concrete in tension.a fictitious crack/crack band model was proposed [46].

For elastic material such as steel.it undergoes huge amount of deformation before it finally gets fractured (Fig.1).Limiting the fracture analysis till the linear material behaviour i.e.use of elasticmoduli in calculating energy release rate may not be exposing the actual physical behavior of structure when it finally undergoes fracture [47].

Fig.1.Engineering stress-strain curve of mild steel [50].

The proposed methodology is capable of solving an inelastic problem in the strength of material style of calculations with reasonable modelling and structural idealization assumptions.It does not involve in exact and precise variation of solution parameters such as the solution by the theory of elasticity or reaching to intermolecular bond for finding the solution but solves the problem with relative ease and can be suitably adopted by designers and engineers.

4.Methodology

As stated earlier.the fundamental approach of structural analysis is essentially equations of static equilibrium-force balance.These equations are differential equations of equilibrium together with boundary conditions.These differential equations together with boundary conditions can also be obtained from the total potential of the structural system.After applying stationarity condition δΠ =0 to the total potential of the system ‘Π’.differential equation together with boundary conditions of the structural problem can be obtained.In this Variational principle.stationarity condition,in essence,is the differentiation of total potential which yields force equilibrium.Finite element analysis follows this Variational procedure.We shall also use the Variational principle to find the relation between the external loads applied with the internal stress generated in the structure in order to formulate the fundamental procedure of the analysis [48,49].

The proposed methodology of making nonlinear analysis essentially lies in locating the point in stress-strain diagram of the material(Fig.1).It can better be explained with the help of equation(10)which is derived in this section.Taking the simple example of the one-dimensional problem of bar of Fig.5 which is loaded axially.Stress generated in the bar can be evaluated if its area of cross-section is known.Hence,by knowing the stress,strain can be found correspondingly by the stress-strain graph of the material.Strain energy can simply be evaluated by simple integration depending upon the type of load and hence,displacement can also be evaluated.

4.1.Effect of large deformation

In the nonlinear range of the material deformation.material deforms quite large and this effect should be accommodated.To accommodate these effects,true stress and true strains are used in the analysis (Figs.2 and 3).Stress and strain functions that are derived from the true stress and true strain can be regarded as the energy conjugates in the proposed methodology.

4.2.Stress and strain functions

The stress-strain curve that is used in this methodology is shown in Fig.1 [50].For convenience.these stress and strain data are mapped to very between-1 and+1,very much like the Natural Coordinate System of a finite element in finite element analysis literature [51-54].Engineering stress and engineering strains are first converted into true stress and true strain by the following relations:

Fig.2.Variation of true stress function along with raw data of true stress in reference coordinate system.

Fig.3.Variation of true strain function along with raw data of true strain in reference coordinate system.

Here.σTand εTare true stress and true strain respectively.Following stress function σ(r)and strain functions ε(r)are derived by the method of curve fitting from the true stress-true strain data of mild steel[55].

Here ‘r′represents the reference coordinate system.Variation of these stress and strain function within the reference coordinate can be observed in Figs.2 and 3.Raw data of true stress-strain is also drawn to observe the accuracy of curve fitting.

4.3.Variational formulation -force equilibrium equation

Einstein notation is used throughout the text of this article[56].Fig.4 shows the three-dimensional structure with traction force Fiacting at surface ΓFtogether with fixed boundary conditions.

Total strain energy (U) generated in the structure because of application of external traction can be calculated as [29],

Fig.4.Typical three-dimensional structure with loading and boundary conditions.

Fig.5.A simple bar loaded axially.

As the structure undergoes some amount of deformation (ui)because of the traction force (Fi) applies at surface ‘ΓF′.External work done can be calculated as,

Hence.total potential of the system can be calculated as,

For developing the force equilibrium equation.consider the example of a simple bar in XY-plane of cross-sectional area ‘A′and length ‘L′which is loaded with traction force ‘F1′in X-direction at the free end (Fig.5).The bar is fixed at another end.The onedimensional bar is shown in Natural coordinates (r.s and t) and Global coordinates(X,Yand Z).Z-axis(t-axis in natural coordinates)can be visualized as perpendicular to the plane of the paper.

As the problem is one dimensional.Axial stress component in xdirection(σ1)only exist in the structural body.As for the loaded bar,stress remains constant at each section along the length.internal strain energy can be evaluated as,

Now.external work done by the external load ‘F1′can be evaluated from Eq.(6) as,

Hence.total potential of the system,

Applying stationarity condition δΠ = 0,

Substituting the value of stress function from equation (3),

This equation is force balance as left side of the equation is simply internal force generated in the bar which is being balanced by external loads.We will abundantly use this equation to find the value of ‘r′in natural coordinate system i.e.locating the point on stress-strain curve of Fig.3.

5.Numerical examples

In this section.problems are solved to demonstrate the methodology.These are solution of a tapered bar.beam bending problem,torsion and energy release rate for sheet with central crack and double cantilever beam (DCB).For each problem results are compared with the classical displacement based linear theory.Modelling and structural idealization is also discussed for each problem in this section.

5.1.Tapered bar

Consider the tapered bar which is fixed at one end and loaded axially at another end as shown in Fig.6.It has sectional area ‘A0′at section(1)and becomes double at section(2).The length of the bar is 5 m.With these geometric properties,if the variation of section is quadratic.then it could be expressed by the following expression[57].

Strain energy stored for this uniaxial case can be computed as,Strain energy

Fig.6.Tapered bar with square variation of cross-section loaded axially.

As area ‘Ax′of the bar varies according to Eq.(10).the strain energy can be modified as,

This integral again can be modified as,

Considering A0as unity and hence area at the free end as 2 m2.Integrating the first integral yields,

Here,r1and r1can be evaluated by Eq.(9)at section(1)and(2)respectively.Substituting stress and strain functions from Eqs.(3)and (4),

This strain energy is evaluated for different loading conditions and the axial deformation of the bar is compared with known solution by linear theory and the results are summarized in Table 1

In Fig.7,results are compared with linear theory where modulus of elasticity is used as material input.From the stress-strain curve,average modulus of elasticity in linear range is calculated as 184128.9683 MPa.From the stress-strain curve of material (Fig.2)yielding starts after approximately 300 MPa.Till the linear limit of the material behaviour.results are in good agreement.After the yield point deformation increases rapidly.It could be interpreted from stress-strain graph of the material.As elastic modulus decreases rapidly as curved flattens after the yield point,deformation of the bar also increases rapidly.

5.2.Beam bending problem

For solving a beam bending problem.structural idealization is considered same as it is with displacement based classical theory.It could be realized that all the linearity assumptions in basic derivation of equations are eliminated.Fig.8 shows two dimensional problem of a cantilever beam of length ‘L′.width ‘b′and depth ‘h′.loaded with a bending moment ‘M′at the free end.

Fig.7.Comparison of results of the tapered bar by the linear analysis approach.

5.2.1.Structural idealization - stress variation in the beam

As the beam is loaded,at any particular section,stresses will be zero at neutral axis and will keep on increasing towards the outer fibers of the beam.either in compression or tension (towards the top or bottom layer).This variation can be glanced in Fig.9 (a).Depending on the magnitude of bending moment.either it generates stresses in linear or nonlinear range of the material behavior,stresses variation across the depth at any particular section will exactly match with the stress-strain variation of the material(Fig.2) and is represented in this Fig.9(a).If the magnitude of bending moment ‘M′increases at the free end.its variation with increasing values of bending moment across the depth at any particular section can be glanced in Fig.9 (b).

For the cantilever beam subjected to bending moment at the free end,bending moment at each section remains constant.If σmaxis the maximum stress that generates at the extreme fiber at any particular section,then the variation of this maximum stress across the length of the beam can be glanced with Fig.10.

5.2.2.Force equilibrium equation

Let Mxbe the bending moment ant any particular section ‘x′of Fig.8.Considering stress ′σx′ in a fiber at any particular depth ‘y′from neutral axis.bending moment at that section can be calculated as,

or,

As stress variation along the thickness direction remains constant and it varies across the depth only,

Fig.8.Simply supported beam with concentrated load at mid span.

Fig.9.(a) Variation of stress across depth of the beam at any particular section.(b).Representation of stress variation across the depth of the beam with increasing magnitude of bending moment.

Fig.10.Maximum stress at the outer layer remains constant along the length of the beam.

Integrating,

Rearranging,

Substituting the stress function from Eq.(3).

It is to be noted that in the derivation,resembles with the moment of inertia of the rectangular sectionwhich is derived by linear theory of classical mechanics.Hereavoids any linearity assumption in derivation.

5.2.3.Strain energy in bending

Strain energy stored in the structural body can be evaluated by,

As the stress distribution across the with ‘b′remains constant at any section,

Consider the stress variation along the depth of any particular section as explained above.strain energy stored either in compression or tension zone only can be evaluated as

Here.lower bond of the integral is -1 as from the stress function.stress is zero at r = - 1.which is at the neutral axis of the beam.Hence this strain energy is evaluated for the half of the beam depth i.e.For the material such as steel,strain energy for the full depth can be evaluated as,

As stress variation along the length of the beam is constant as represented by Fig.10,

Substituting stress and strain functions from Eqs.(3) and (4),

Table 1 Results of the tapered bar with quadratic variation.

Table 2 Results summary of the cantilever beam for different magnitudes of load.

Fig.11.Bending energy stored in the beam for different magnitudes of loads.

Considering the width ‘b′of the beam as 30 mm,depth of each cantilever ‘h′as 15 mm and length of the beam as 1 m.If the beam is loaded with the bending moment of ‘M′the free end.value of ‘r′corresponding to maximum stress at the extreme fiber can be evaluated by Eq.(14.a).Once value of ‘r′in reference coordinate is found,strain energy can be evaluated by Eq.(15.a).Rotation at the free end of the cantilever beam can also be evaluated by the following expression.

Fig.12.Rotation at free end of beam for different magnitudes of loads.

Analysis is repeated for different magnitudes of bending moment ‘M′and the results are summarized in Table 2.

These tabular results can be visualized and compared with the help of the following Figs.11 and 12.

It is worthy to note that for the bending moment of 600 N·m and more.stresses enter into nonlinear range of the material behavior.As with the material behavior.strain energy as well as rotation increases tremendously after the yield point (Fig.2).

Fig.13.Geometric dimensions of circular shaft tested for torsion test [58].

5.3.Torsion problem

For solving torsion problem.steel specimen containing 0.1%carbon and geometric dimensions shown in Fig.13 was tested in the lab [58].During the experiment.specimen experienced maximum torque of 20.72 N-m at the rotation angle of 2.0584 radians.The specimen gets fractured after the rotation angle of 34.8889 radians.

In this methodology,shear stress is evaluated by Eq.(22)which is derived in section 5.3.2 of this article and shear strains are evaluated by the following expressions.

Fig.15.Specimen after fracture in torsion.

Fig.16.Circular Shaft with the couple acting at the free end.

In the torsion experiment.we could read the total torque experienced by the specimen,which was at the outer periphery of the section and is the maximum one.The following shear stress and shear strain functions are derived from the experimental data with the method of curve fitting [55].

Variation of shear stress function with experimental data can be observed in Fig.14.

Fig.17.Stress distribution in the circular section during the load in nonlinear range.

Fig.18.Stress variation along the radius.

It should be noticed that stress and strain functions are not mapper in reference coordinate system for solving the torsion problem.During the lab experiment.it is observed that the area/volume of cross-section does not change considerably when fractured with a torsional load (Fig.15).It is unlike the deformation pattern when the specimen is fractured while loaded axially.

5.3.1.Structural idealization-stress variation in the shaft

Fig.16 shows a circular shaft which is fixed at one end and loaded with a torsional force ‘T′at the free end.If the magnitude of torsion is such that the stress enters in nonlinear range of the material behavior,then stress distribution along the length as well as along the section will be varying exactly as represented in Figs.16 and 17.

Stresses will be zero at the center and will keep on increasing towards the outer radius.Here it is reasonable to assume the modelling and structural idealization as with the conventional displacement based for torsion i.e.material is homogeneous.circular section remains circular and remains plane after deformation,the cross-section is rigid in rotation,each point in diameter rotates through the same angle etc.For the results comparison purpose,we idealized the structure with the same assumptions [59-61].

5.3.2.Force equilibrium equation

External torque applied on the shaft generates internal shear stresses varying across the radius as shown in Fig.18.In this section,total torque is evaluated which is generated by internal stress distribution along the section.

Considering the strip of width ‘dρ′at radius‘ρ′from the center at which the magnitude of stress is τ′(Fig.18).Force associated with this strip can be evaluated as,

Torque generated because of this force can be calculated as

If the magnitude of torque associated with this thin strip is T′,T′= τ × 2πρ×dρ × ρ

Substituting T′= F × ρ

Total torque generated because of this stress distribution can be evaluated by simple integration throughout the sectional area as,

where ‘R′is the radius of the circular shaft.This equation cannot be integrated along the radius as shear stress also varies along the radius as shown in Figs.16 and 17.Differentiating the equation both sides,

Multiplying both sides by ‘ρ′,

At the outer surface,

Or

This expression gives the value of maximum shear stress ‘τ′generated at the outer periphery of the circular shaft.By simply substituting the value of torque ‘T′.maximum shear stress generated at the outer periphery can be evaluated.Now.substituting shear stress function from equation (18) into this equation (22).

For the known value of torque on the shaft.total angular rotation ‘θ′can be evaluated by this equation.It is worthy to note that the stress function of Eq.(18)is the link between stress generated in the structure to the angle of twist.This expression works like elastic moduli of the linear elastic method.The expression gives ideal results in linear as well as in the nonlinear range of the material behavior.

5.3.3.Strain energy in torsion

Strain energy in the circular shaft can now be evaluated by,

As shear stress remains same at any particular cross section along the length.hence,

Considering the strip of width ‘dρ′at a radial distance ‘ρ′from the center of Fig.8.strain energy can be evaluated throughout the section by,

Table 3 Results for the circular shaft loaded in torsion.

Substituting stress function ‘τ′.strain function ‘γ′along with geometrical dimensions of the shaft.

This equation can be used for evaluating strain energy stored in the circular shaft of any typical geometric dimensions.For the given material,if stress and strain functions such as Eqs.(18)and(19)are derived once.by simply substituting geometric dimensions along with stress and strain functions,strain energy can be evaluated by this expression.Taking length of the circular shaft as 1 m and radius 10 mm.analysis is repeated for different magnitudes of torque at fee end and the results are summarized in Table 3.

These results can graphically be viewed with the help of Fig.19.

For the results comparison purpose,average modulus of rigidity(G) is evaluated as 47072.34 GPa from the linear range of material behavior.The results from the methodology of Fig.19 shows close agreement with the accurate results.

5.4.Sheet with central crack

Considering a metal strip of thickness ‘b′.width ‘w′and length‘L′.It is fixed at one support and stretched along the length ‘L′by constant stress ‘σy′as shown in Fig.20.

Strain energy stored in a stressed body can be calculated as[53],

Since stress remains constant at each cross-section along the length,

where ‘A′is the area of cross-section of the strip (A =b*w).Since the sheet is loaded with the constant load.stress ′σ′ and corresponding strain ′ε′remains constant at any particular cross section for the given loading condition.So.the strain energy can be computed as,

Fig.19.Variation of strain energy for different magnitudes of torsion force.

If the central cut is made in the strip of crack length ‘2a′(Fig.20),energy released from this portion of the sheet can be evaluated as,Energy release=Volume of the two triangles×Strain energy density

Fig.20.Metal sheet with central crack loaded longitudinally-(a)undeformed state,(b)deformed state.

Table 4 Result analysis of plate with central crack.

Or,

In this expression.it is assumed that if a central cut is made(perpendicular to the direction of stress)of length ‘2a′in the sheet then the portion where energy is released is triangular shape and its height ‘λ2a′.Where λ.It is exactly same assumption that is considered in the literature for this problem [34,35].Substituting the values of ‘σ’and ‘ε’ of the mild steel from Eqs.(3) and (4),

Here.the value of ‘r′can be evaluated by Eq.(10).Now.by the definition of energy release rate (G),

where ‘Π′is total potential of the system and ‘A′is newly created area by the advancement of crack.Hence,energy release rate can be computed as,

Simplifying,

Fig.21.Energy release rate for the plane sheet with central crack.

Considering thickness ‘b′of the sheet as 10 mm.width ‘w′as 50 mm and length ‘L′as 100 mm and initial crack length ‘a′as 5 mm,energy release rate is computed for different values of σyand are summarized in Table 4.These results are also compared with the energy release rate by classical fracture mechanics.

Fig.21 shows the variation of energy release rate versus the stress(σy)applied at the one end of the sheet.It is to be noticed that energy release rate increases rapidly after the stress of 300 MPa.It is actually the yield point(at 0.2%strain)at the stress-strain curve of the material adopted(Fig.2).

Fig.21 shows the numerical comparison of the energy release rate evaluated by this methodology versus the classical linear theory.Results show the close agreement till the moment loading is in linear range of the material behaviour.

5.5.Double cantilever beam specimen

Consider the double cantilever beam(DCB)specimen of Fig.22.It has two cantilever arms,each with width ‘b’and depth ‘h’.It has initial crack length of ‘a’.

As the DCB has two cantilever arms loaded with bending moment at free end.strain energy stored can be evaluated by equation (15),

Now,energy release rate for DCB specimen can be evaluated by,

Fig.22.Double cantilever beam (DCB) specimen.

Substituting the values,

Or,

Substituting the stress and strain functions from Eqs.(3)and(4),

Considering the width of the DCB specimen ‘b′as 30 mm,depth of each cantilever‘h′as 15 mm and initial crack length ‘a′as 50 mm,the energy release rate for different loading conditions is evaluated and the results are summarized in Table 5.

Fig.23 shows the variation of the energy release rate of the DCB specimen versus the maximum stress generated at the end of the cantilever arm.These results are compared with the classical linear theory in the literature.It can be observed as the energy release rate increases rapidly after the stress of 300 MPa which is actually yield point of the material adopted in the analysis(Fig.2).

Fig.24 shows the result comparison of energy release rate of DCB specimen and plate with central crack.As energy release rate is computed for two different structures,it shows in this comparison that energy release rate is the material property.

Fig.23.Energy Release rate for DCB specimen.

6.Conclusions

Observing the solutions of all problems.it can be concluded as the methodology is capable of giving excellent results in linear as well as in nonlinear range of the material behavior.It does not need any repetition and then convergence procedure as with the classical nonlinear analysis.

Fig.24.Comparison of Energy release rate of DCB specimen and plate with edge crack.

Table 5 Results analysis of DCB specimen.

From the solutions of basic problems such as tapered bar,beam bending and torsion problem,it is noted that this methodology has given birth to new formulas and equations which eliminates all the linearity assumptions in their derivations.With this.it also eliminates any possibility of generation of error in the solution of the problem.

From the solutions of energy release rate from plate with central crack and DCB specimen.it can be concluded that results are in close agreement with that of the classical theory (Figs.21 and 23)till the moment loading on structure is in linear range of the material behavior.The methodology is capable of exposing structural behaviour as the stress enters in the nonlinear range of the material behaviour till it finally fractures.The analysis procedure gives much more clearer picture of plastic/fracture behaviour of the problem at hand.

In the fracture mechanics literature.failure of the material is considered when the load reaches to the yield point [34,35].From the stress stain graph of Fig.1 it can be observed as the yielding starts at(at 0.2%strain)300 MPa of stress.Graphs of Figs.21 and 23 shows the variation of the energy release rate versus the stress generated in both structures.In these figures(Figs.21 and 23),the energy release rate increases rapidly after 300 MPa.This shows the material behaviour enters in nonlinear range according to the load increment.

The analysis procedure is simple to adopt.The methodology solves the fracture mechanics of plasticity problems in a much more easy way with greater insight of the physical problem.It solves the fracture mechanics problem in the strength of material style with relative ease without being involved in very accurate solution such as by theory of elasticity.Hence.it can conveniently be adopted for engineering and design purposes with relative ease.

The method is clear in fundamental with better physical infight of the structural response in plastic range and it is much more accurate than the displacement based linear methods that we use traditionally.

Declaration of competing interest

The author(s) declared no potential conflicts of interest with respect to the research.authorship and publication of this article.