towards simulation of elasto-plastic deformation: an investigation

Sadhana Vol. 27, Part 3, June 2002, pp. 251–294. © Printed in India

Towards simulation of elasto-plastic deformation: Aninvestigation

ARUN R RAO and U SHRINIVASA

Department of Mechanical Engineering, Indian Institute of Science,Bangalore 560 012, Indiae-mail: [email protected]

MS received 5 December 2001

Abstract. Can the deformation of a solid body during plastic flow be assumed tobe similar to that of fluids? Here we investigate the possibility of using a modifiedNavier–Stokes equation as the governing differential equation by including elasticresistance. We adopt the microscopic point of view to explain the material behaviourby laying special emphasis on strain localisation and tension instabilities. A springand damper model is constructed to obtain approximate simulation of the materialbehaviour. Based upon the understanding developed from simulating simple tests,we re-formulate the field equation using resistances to change in volume and shape.The new field equation reduces to the Navier–Stokes equation in the fluid limit andCauchy’s equation in the solid limit. The viscosity and second viscosity of fluidsare clearly defined. Bulk and shear modulii and solid damping determine the solidbehaviour. Pressure disappears from the field equation and so there is no need toinvoke the continuity equation. The four material parameters are determinable fromsimple measurments. This paper tries to capture the various steps of the investigationwhich lead to the final result.

Keywords. Elasto-plastic deformation; modified Navier–Stokes equation;strain localisation; tension instability; material behaviour; Cauchy’s equation.

1. Introduction

The mathematical theory of plasticity has been in use for many years but the formulation ofa field equation for simulation of elasto-plastic deformation is still a matter of discussion.Several engineering idealised theories are proposed but they fail to account for the variousphenomena observed. Due to the complicated nature of the plasticity problem, theory in earlierdays is chiefly confined to small deformation, viz. infinitesimal theory with some specialemphasis on hardening laws and, some special material responses (Hill 1950; Prager & Hodge1951; Prager 1955; Drucker 1960; Naghdi 1960). These theories are ‘rate-independent’ tocover some general class of materials with consideration of a few special properties such asmaterial anisotropy, creep and thermal effects.

A large amount of work is carried out by different schools of plasticity in the area of finitedeformation either in the Lagrangian or in the Eulerian coordinate systems using stress-based

251

252 Arun R Rao and U Shrinivasa

or strain-based approaches. The stress-based formulation (Green & Naghdi 1965, 1966) isin turn found inadequate (Naghdi & Trapping 1975) to explain some material properties andis non-trivial for implementing numerically. Later Naghdi and co-workers developed strain-based formulation (Naghdi & Trapp 1975; Casey & Naghdi 1981, 1983) to overcome thesedifficulties and made it suitable to solve numerically.

In the literature, we can often see the development of the constitutive equations for defor-mation theories on macroscopic scale (Barbeet al 2001). But the experimental observationsmade by Taylor & Elam (1923, 1925, 1926) reveals the importance of microscopic structureduring large deformation. These observations led further to development of various micro-scopic theories (Batdorf & Budiansky 1949; Koiter 1953; Kroner 1960; Budiansky & Wu1962; Hill 1965, 1966; Hutchinson 1970; Rice 1976; Bazant 1984) and more recently tothose of Qui & Weng (1992, 1995), Rubin (1996), Acharya & Beaudoin (2000), Bigonietal (2000), and Acharya (2001). All these workers tried to develop the theory by consideringvarious aspects of microstructure like single crystal or polycrystal deformation, strain local-isation, material anisotropy etc. Nowadays, these microstructure theories are covering moreand more ground as the physics of micro-mechanics is being deeply studied and understood(Bertram 1998) to provide answers to most of the questions.

With the development of theory, efforts are made to develop the plasticity theory in theEulerian coordinate system as it is found more suitable for large deformations. Kroner (1960)initiates work in Eulerian coordinate system and later it is carried out by the others (Backman1964; Lee & Liu 1967; Lee 1969; Dienes 1979, 1986; Leeet al 1983; Loret 1983; Nemat-Nasser 1983; Dafalias 1983, 1984; Atluri 1984; Agah-Tehraniet al 1987). In recent yearswe see some schools of plasticity use the ‘arbitrary Lagrangian-Eulerian’ description, whichcombines both Lagrangian and Eulerian descriptions to solve the problems (Haber 1984;Huetinket al1990; Ghosh & Suresh 1996; Braess & Wriggers, 2000).

Most of the existing plasticity theories depend heavily upon yield function to differentiatebetween elastic and plastic regions (Hill 1994). Yield function is dependent upon stress, plasticstrain and work hardening parameters and is usually determined by experiments. However,the existence of the yield surface or yield condition is questionable. Some investigators (Seth1963, 1964; Valanis 1971, 1975; Bodner & Partan 1975; Lee & Zaverl 1978, 1979; Stouffer& Bodner 1979) lay special emphasis on the possibility of developing the theory of plasticitywithout introducing a yield surface or yield condition.

A body is understood to deform simultaneously by elastic and plastic strains under theapplication of external load. In the classical theory of elasticity the linear trend of infinites-imal deformation is expressed by the generalised Hooke’s law. In actuality, the cyclic load-deformation or stress–strain curve is not a single-valued function but forms a hysteretic loop(Lazan 1968). Lazan (1968) also observed that all materials, composites and structural assem-blies do not behave in a perfectly elastic manner even at very low stresses. Inelasticity isalways present under all types of loading. Hence the constitutive equations for the elastic andthe plastic parts of the deformation cannot be treated independently. Elastic and plastic effectsshould dovetail into each other even when elastic behaviour is assumed to be unaffected bythe preceding plastic deformations. This supposition avoids any assumption of yield condi-tion or yield surface to determine any elastic–plastic boundary, which may or may not exist(Seth 1970) and is sometime also ambiguous in multi-axial cases (Chen & Lu 2000).

Existing plasticity theory is based on certain experimental observations on the macroscopicbehaviour of metals in the uniform state of combined stresses (Chakrabarty 1998). Theseobserved results are then idealised into a mathematical formulation to describe the behaviourof metals under complex stresses. However a lot of assumptions are involved to simplify the

Simulation of elasto-plastic deformation 253

complex phenomenon of plastic deformation. These assumptions make the theory incapableof explaining various material phenomenon such as the Bauschinger effect, anisotropy, timeand temperature effects, size effect, stress inhomogeneity, material damping etc.

Thus, existing models are able to explain some of the phenomenon observed, but leavebehind a lot of controversies. Each model is useful for a particular purpose with some well-defined boundary conditions. However, many phenomena occurring during plastic deforma-tion remain unexplained. Hence, there is an urgent need for a constitutive equation whichin general will correctly explain deformation behaviour and also the physics involved in thedeformation process. Therefore we try to develop a field equation from first principles (asagainst from experimental observations) which assumes a generalised relation between stress,strain and strain rate that could represent the material behaviour in the elastic range, transi-tion from the elastic to plastic state and in the plastic range, and later use the same to explainmost of the phenomena observed in practice.

2. Formulation

Zienkiewicz & Godbole (1975) pointed out that in metals, plasticity with large deformation,whose elastic strains are negligible, stands on the borderline of fluid and solid mechanics.Under equilibrium conditions both solid and fluid are identical provided dynamic forces areneglected. Thus one can conclude that the solution of a viscous flow problem is identicalto the solution of an equivalent incompressible elastic problem in which displacements areinterpreted as velocities and infinitesimal strain as strain rate (Zienkiewicz & Godbole 1974).Thus all the techniques available in the literature for the solution of one problem are applicableto the other. The analogy that in the plastic state metals behave like viscous fluids helps toexpress the formulation in the Eulerian coordinate system which seems more relevant in largeelastic–plastic deformation (Prager 1961). Further we see that in recent and current literature,most plasticity theories are inclined towards an Eulerian formulation in stress space setting.This is mainly because of its analogy with viscous fluid flow, the construction of the theoryin terms of Cauchy stresses and its rate being more fundamental. Cauchy’s stress tensorP isdefined in the current configuration as,

P = τij ei ⊗ ej

whereτij are the stress components andei andej are the unit co-ordinate vectors.

2.1 Contribution from the strain-rate tensor

Every material displays more or less viscous properties. For many important structural mate-rials, these properties appear to surface after the plastic state has been reached. It is normallybelieved that materials display viscous properties only in the plastic range. We however lookfor a universal model.

Consider the stress (following Lamb 1945) exerted across any small area situated at a pointO in the continuum (figure 1). The deviation of the state of stress denoted bypxx, pxy, . . .

from one of pressure uniform in all directions depends entirely on the motion of distortion(i.e. on instantaneous rate of deformation) in the neighbourhood of O, i.e. on 6 quantitiesa, b, c, f, g, h which are defined as,

a = ∂u/∂x, b =∂v/∂y, c =∂w/∂z,

f = ∂w

∂y+ ∂v

∂z, g =∂u

∂z+ ∂w

∂x, h =∂v

∂x+ ∂u

∂y, (1)


O

z’

p1

y’

p3

p

x’

p2

Figure 1. Force equilibrium for a point in fluid.

and, the rate of the deformation matrixA, turns out to be

A = a h/2 g/2

h/2 b f/2g/2 f/2 c

. (2)

We neglect here the rotation tensor because the continuum rotation due to deformation is realbut irrelevant as it does not contribute to stresses (Drucker 1988). In other words, “the plasticspin nullifies the continuum spin” (Dafalias 1985).

Let a′, b′, c′ be the rates of extension along principal axes of distortion. Letp1, p2, p3 bethe principal stresses exerted at O across the planesy ′z′, z′x ′, x ′y ′ and wholly perpendicularto these planes andp be the mean stress, which is the arithmetic mean of the normal pressureon any three mutually perpendicular planes through the point O.

Now p1, p2, p3 differ from −p by quantities depending on the matrix of distortion,which must be functions ofa′, b′, c′ only. The hypothesis made in the classical theory isthat these functions are linear, i.e. they depend on the first invariant of the rate of deformationtensor (Lamb 1945). Considering however the remaining two invariants also of the sametensor, we can write (following Anjaneyulu 1999),

p1 = −p + λ1(a′ + b′ + c′) + λ2a

′ + λ3a′b′ + λ4(a

′b′ + b′c′ + c′a′) + λ5a′b′c′,

p2 = −p + λ1(a′ + b′ + c′) + λ2b

′ + λ3b′c′ + λ4(a

′b′ + b′c′ + c′a′) + λ5a′b′c′,

p3 = −p + λ1(a′ + b′ + c′) + λ2c

′ + λ3c′a′ + λ4(a

′b′ + b′c′ + c′a′) + λ5a′b′c′.

(3)

Adding these three equations and using the definition ofp, i.e.,p = (p1 + p2 + p3)/3 wecan solve for the constants. Thus the resistance part due top and the strain-rate tensor in the


principal coordinates become,

P′d = −pI + λ1J

′1I − 3λ1

a′ 0 0

0 b′ 00 0 c′

− 3λ4J

′3

a′ 0 0

0 b′ 00 0 c′

−1

+ λ4J′2I,

(4)

whereJ ′1, J ′

2 andJ ′3 are the first, second and third invariants of strain rate tensors in the

principal coordinates system. By transformingP′d in principal coordinates to the current

coordinates, we get,

Pd = LT P ′dL

= −pI + λ1(J1I − 3A) + λ4(J2I − 3Adj (A)). (5)

Hereλ1 andλ4 are the unknown material constants andJ1, J2 are the first two invariants ofthe strain rate tensor in the current coordinate system.

2.2 Contribution from the strain tensor

In classical elasticity theory, stresses are assumed to be linear functions of strains, and aregiven by the generalised Hooke’s law. Anjaneyulu (1999) considers all the three invariants ofthe strain tensor, to obtain a generalised relation as,

Pe = λI1I + 2µE + η1Adj (E) + η2I2I + η3I3I, (6)

whereE is the small strain tensor,

E = εxx εxy εxz

εyx εyy εyz

εzx εzy εzz

,

εxx = ∂u/∂x, εyy =∂v/∂y, εzz =∂w/∂z,

εxy = εyx = ∂u

∂y+ ∂v

∂x, εyz =εzy = ∂v

∂z+ ∂w

∂y, εzx =εxz = ∂w

∂x+ ∂u

∂z,

(7)

whereλ andµ are Lame’s constants,η1, η2, η3 are unknown material constants andI1, I2

andI3 are the first, second and third invariants of strain in the current coordinate system. Sinceplastic flow is usually associated with large deformation gradients, we consider it appropriateto use the Almansi strain tensor, which in spatial form is given by,

E∗ij = 1

2

[∂ui

∂xj

+ ∂uj

∂xi

− ∂uk

∂xi

∂uk

∂xj

]. (8)

The above strain tensorE∗ is an exact one in the Eulerian coordinate system and no approx-imations are involved (Fung 1965). While using the expression forPe there is no explicitreference to the original undeformed state. We need only the gradients ofPe to set up theequations of motions in the next section.

It is commonly accepted in practice that, during elasto-plastic flow, instantaneous unloadingis always linearly elastic to begin with. This is difficult ifη1, η2 andη3 have non-zero


values. Therefore to be consistent with observations particularly on steels, we assume thatthe constantη1 = η2 = η3 = 0, which reduces the expression forPe to,

Pe = λK1I + 2µE∗, (9)

whereK1 = E∗kk.

2.3 Final form of the equations of motion

The final form of the equation of motion can be obtained by substituting the contributionsfrom strain and strain rate tensors in the equation for the law of balance of linear momentumin spatial (Eulerian) form (Malvern 1969),

ρDUDt

= ρB + (∇ · P T )T , (10)

where,B = [Bx By Bz

]Tis body force per unit mass alongx, y and z axes respectively,

ρ = ρ(x, t) is the mass density,U are velocities inx, y and z directions which providethe Eulerian form of reference andP the Cauchy stress tensor, which is defined in currentconfiguration. To obtain the complete expression for stress tensor, the two tensorsPd andPe

which correspond to the non-elastic and elastic parts respectively are added, i.e.P = Pd +Pe.Substituting the expressions obtained earlier, we get,

ρDUDt

= ρB + ∇T (−p + λK1 + λ1J1 + λ4J2) + [∇ · (2µE∗ − 3λ1A

− 3λ4Adj (A))]T . (11)

In the above equation of motion,λ1 is similar to the classical fluid viscosity andλ4 is anunknown nonlinear viscosity coefficient representing additional resistance to advection insolids. There is a new state variable called pressure,p, which is only a computationalintermediary and agrees with classical fluid pressure when elastic resistance is not there.This variable is equivalent to mean stress in solids (Zienkiewicz & Godbole, 1975). Thereare four dependent variables, the three components of the displacement vector, i.e.,u, v, w

and a pressure quantityp, all of which are functions of space variables and time. Thereforeto generate the additional equation required, the equation of motion has to be supplementedwith the continuity equation.

The law of conservation of mass i.e. continuity equation in spatial (Eulerian) form (Malvern1969) is given by,

Dρ/Dt + ρ(∇ · U) = 0 (12)

whereρ = ρ(x, t) is the mass density andU the velocities inx, y andz directions.For convenience in the later part of the paper we will express here the mass densityρ in

the form of volumetric strainυ as,

D

Dt

(1

υ

)+ 1

υ(∇ · U) = 0. (13)

The 1-D and 2-D forms of the new field equation in cartesian coordinates and in cylindricalcoordinate system reduces to the same form considering the strain in ther andz directions.But the second unknown material constantλ4 does not appear in this equation. Thus, eventhough it is believed that the 2-D cylindrical form in terms ofr andz is the reduced 3-Dcartesian coordinate form, it does not help us to obtain the importance of the second unknownmaterial constantλ4 which appears only in 3-D.


2.4 Dimensionless form of the equation of motion

So far the equations of motion developed are complex, difficult to interpret both geometricallyand physically, and to solve analytically if they are kept in the same form. These equationsshould be put into compact forms in order to compare their behaviour with experimentalobservations. Hence we carry out nondimensionalisation of the equations. This also helps(White 1994) to reduce the number and the complexity of the variables.

The dimensionless form of the 1-D equation of motion is obtained by nondimensionalisingthe terms as follows,

u = (u/u0) u0 = uu0 ⇒ du = u0du,

t = (t/t0) t0 = t t0 ⇒ dt = t0dt ,

x = (x/x0) x0 = xx0 ⇒ dx = x0dx,

p = (p/p0) p0 = pp0,

ρ = (ρ/ρ0) ρ0 = ρρ0,

Bx = (Bx/Bx0) Bx0 = BxBx0,

u = du

dt= u0du

t0dt= u0

t0˙u,

du

dx= u0du

x0dx,

du

dt= u0d ˙u

t0t0dt= u0d ˙u

t20dt

,

du

dx= u0d ˙u

t0x0dx,

where,u0, t0, x0 . . . etc. are the known scale factors andu, t , x . . . etc. are the correspondingnondimensionalised variables. Thus the 1-D form of the nondimensionalised field equationbecomes,

d ˙udt

+ ε0 ˙ud ˙udx

= ζ Bx + υd

dx

[−βp − 2γ

d ˙udx

+ 1

2χ

{2

du

dx− ε0

(du

dx

)2}]

,

(14)

where,

ε0 = u0/x0,

ζ = Bx0t20/u0,

υ = 1 + υ1 = 1 + (u0du/x0dx),

β = p0t20/(ρ0u0x0),

γ = λ1t0/(p0x20),

χ = (λ + 2µ)(t20/ρ0x

20),

are the dimensionless constants. We can writeβ as,

β = p0t20

ρ0u0x0= p0t

20

ρ0x20 (u0/x0)

.


Let us define,

p0 = (u0/x0) (λ + 2µ). (15)

Then we get,

β = (u0/x0) (λ + 2µ)t20

ρ0x20 (u0/x0)

= (λ + 2µ)t20

ρ0x20

= χ.

Now we can writeχ as,

χ = (λ + 2µ)t20

ρ0 (u0/x0)2 u2

0

,

where (u0/x0)2 is the limiting strain during elasto-plastic loading say ‘ultimate strain’

whereas,u0/t0 = u0 is the rate of deformation, i.e. the velocity at which load is applied.Further we can see that,

(λ + 2µ)/ρ0 ∝ E/ρ = c2,

wherec is the 1-D wave velocity in the metal, i.e.

χ = β = %c2

u20

(u0

x0

)2

,

where% is a constant.From the dimensionless form we can see that 1/γ has a form similar to that of Reynolds

number Re in the case of fluids. Thereforeλ1 must be a material constant corresponding to thematerial viscosity term. Thus the 1-D equation of motion developed reduces to the simplifiedform

d ˙udt

+ ε0 ˙ud ˙udx

= ζ Bx + υd

dx

[β

(−p + du

dx− 1

2ε0

(du

dx

)2)

− 21

Re

d ˙udx

]. (16)

We can see that, during large deformation, the higher order terms of the equation play animportant role.

A similar procedure is carried out for 2-D and 3-D equations of motion to get the dimen-sionless forms. We consider some characteristic lengthl and a characteristic increment inlength1. The final form of the dimensionless 2-D equation of motion becomes,{

C1

C2

}= ζ

{Bx

By

}+ υ

{∂/∂x

∂/∂y

}(−βp + χ ′K1 + γ J1)

+ υ

[∂

∂x

∂

∂y

] [χ ′′[

Exx Exy

Eyx Eyy

]− 3γ

[a h/2

h/2 b

]],


where,

ζ = gt20/1, β = p0t

20/ρ0l1, γ = λ1t0/ρ0l

2,

χ ′ = λt20/ρ0l

2, χ ′′ = µt20/ρ0l

2,

υ = 1 + ε0

(∂u

∂x+ ∂v

∂y+ ε0

∂u

∂x

∂v

∂y

),

C1 = ∂ ˙u∂t

+ ε0

(˙u∂ ˙u∂x

+ ˙v ∂ ˙u∂y

),

C2 = ∂ ˙v∂t

+ ε0

(˙u∂ ˙v∂x

+ ˙v ∂ ˙v∂y

),

K1 = Kxx + Kyy,

Kxx = Exx = ∂u

∂x− ε0

2

((∂u

∂x

)2

+(

∂v

∂x

)2)

,

Kyy = Eyy = ∂v

∂y− ε0

2

((∂u

∂y

)2

+(

∂v

∂y

)2)

,

Exy = ∂u

∂y+ ∂v

∂x− ε0

(∂u

∂x

∂u

∂y+ ∂v

∂x

∂v

∂y

),

a = ∂ ˙u/∂x, b = ∂ ˙v/∂y, h = ∂ ˙v/∂x + ∂ ˙u/∂y.

The dimensionless form of the 3-D equation becomes,

∂ ˙U/∂t + ε0˙U · ∇ ˙U = ζ B + υ∇T (−βp + χ ′K1 + γ J1 + ηε0J2)

+ υ∇ · [χ ′′E − 3γA − 3ηε0Adj (A)], (17)

where,

K = Kxx + Kyy + Kzz,

Kxx = Exx = ∂u

∂x− ε0

2

((∂u

∂x

)2

+(

∂v

∂x

)2

+(

∂w

∂x

)2)

,

Kyy = Eyy = ∂v

∂y− ε0

2

((∂u

∂y

)2

+(

∂v

∂y

)2

+(

∂w

∂y

)2)

,

Kzz = Ezz = ∂w

∂z− ε0

2

((∂u

∂z

)2

+(

∂v

∂z

)2

+(

∂w

∂z

)2)

,

Exy = ∂u

∂y+ ∂v

∂x− ε0

(∂u

∂x

∂u

∂y+ ∂v

∂x

∂v

∂y+ ∂w

∂x

∂w

∂y

),

Eyz = ∂v

∂z+ ∂w

∂y− ε0

(∂u

∂y

∂u

∂z+ ∂v

∂y

∂v

∂z+ ∂w

∂y

∂w

∂z

),

Ezx = ∂w

∂x+ ∂u

∂z− ε0

(∂u

∂z

∂u

∂x+ ∂v

∂z

∂v

∂x+ ∂w

∂z

∂w

∂x

),

a = ∂ ˙u/∂x, b = ∂ ˙v/∂y, c = ∂ ˙w/∂z,


f = ∂ ˙w∂y

+ ∂ ˙v∂z

, g = ∂ ˙u∂z

+ ∂ ˙w∂x

, h = ∂ ˙v∂x

+ ∂ ˙u∂y

,

J1 = a + b + c,

J2 = ab + bc + ca − 14(f 2 + g2 + h2),

η is the additional dimensional constant due to the unknown material constantλ4 whichappears only in 3-D,

η = λ4/ρ0l2.

Similarly, we can write the dimensionless form of the continuity equation as,

0 = ∂

∂t

(1

υ

)+ ε0

(˙u ∂

∂x

(1

υ

)+ ˙v ∂

∂y

(1

υ

)+ ˙w ∂

∂z

(1

υ

))

+ ε0

υ

(∂ ˙u∂x

+ ∂ ˙v∂y

+ ∂ ˙w∂z

). (18)

3. Model representation and solutions

The governing differential equations developed in the last section represent the elasto-plasticdeformation considering the effect of strain and strain rate. These equations consists of fourmaterial constants, viz.λ andµ which are Lame’s constants,λ1 which appears as a materialviscosity term, andλ4 which is an unknown material constant. The new material constantλ4 is associated with the second-order term in the governing differential equation and fromdimensionless analysis it shown that it has the dimension of kg/m with dimensionless constantη. In the literature there is no mention of such a constant and any corresponding data. In thissection we look into the various material phenomena which are observed during deformationof materials under loading. Since we do not have any data onη, we concentrate on theremaining resistance terms and investigate load deflection behaviour of simple models. Linearelastic resistance can be modelled with springs and nonlinear strain displacement behaviourcan also be modelled by considering nonlinear incremental analysis. Dampers do representviscous dissipation and therefore we begin by considering solids represented by grillages ofsprings and dampers and use of incremental loading. Our main focus is on mild steel of whichinclusions and other nonhomogeneities are an integral part.

In the newly developed governing equations we see that for the usual tension test we canneglect acceleration terms and body force terms as they are very low while compared to theapplied load. Then the equation is governed by the gradients of strain and strain rate. Thestrain rate terms appear in the form of viscous resistance which is synonymous to materialdamping. To represent and to interpret the complex phenomenon occurring inside the material,we treat the material composed of a very large number of particles interconnected by springs(Timoshenko & Goodier 1970). We treat these hypothetical spring elements in parallel witha damper (figure 2) to represent the effect of momentum transfer between the particles.

Each of these combinations of spring and damper, arranged in the form of a cubical structurewith diagonal and body diagonal elements as shown in figure 2, represent in an approximateway many aspects of the resistance of a material to deformation and its directional behaviourunder load. These cubes are arranged side by side (figure 3) to represent the complete tension


F

F

(a) (b)

Figure 2. Spring and damper model.(a)A complete cube with elements.(b) Eachelement in a cube.

test specimen in length, width and depth. This whole specimen can be treated as a homoge-neous material.

To start the computation, we consider a standard specimen size for the tension test prescribedin ASTM (1985) as shown in figure 4. we apply simple tension on this test piece consideringonly the gauge length.

However but the results of the analysis given in figure 5 show that the model does not exhibitthe expected behaviour. It shows stiffening rather than the softening behaviour observed inpractice, i.e. in the experimentally observed stress–strain curve for the tension test. In onesense this is expected as we use a spring element which has the characteristic behaviour ofstiffening during large load.

Thus the strain and strain rate terms appearing in the governing differential equation arenot enough to catch the complete material behaviour. There should be some other factor apartfrom the governing differential equation causes large deformation under high load. Hence welook into the effect of inhomogeneities present in the materials. In the literature we can see

z

xy

F

F Figure 3. Tension test model.


All dimensions are in mm

7522575

50 40

13 R

200

450

5(min)

PLATE TYPE

Figure 4. The standard specimen.

that much theoretical and experimental work has been carried over on these aspect leading tothe microscopic theory of plasticity, which attributes the plastic deformation to microscopicinhomogeneities present in the material. Therefore we try to model the microscopic inhomo-geneities as explained in the next section.

3.1 Effect of microscopic inhomogeneities

From the literature we can see that microscopic discontinuities in the material due to inho-mogeneities result in nonuniform stress distribution in their vicinity. Stresses in these regionsare higher than the average stress in the material. Values of stress concentration depend uponthe size, shape and orientation of the inhomogeneity. For example, a very narrow inclusion(elongated ellipsoid) normal to the direction of force results in very high stress concentrationas compared to one within the direction of force.

The inhomogeneities in our model are created by changing stiffness values of some ran-domly selected springs to very low values as compared to the stiffness of other springs. Due tosudden changes in stiffness values, these springs act like soft inclusions in the metal matrix.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

50

100

150

200

250

300

350

z displacement

Incr

emen

tal l

oad

(nor

mal

ised

to 0

.2%

str

ain)

Total deformationFigure 5. Effect of damping on load-deformation behaviour.


0 10 20 30 40 50 60 70 80 90 100−100

−50

0

50

% number of elements

% c

hang

e in

forc

e du

e to

inho

mog

enei

ties

Compression range

Tension range

Material with 1% inhomogeneity




Figure 6. Comparison of mate-rial containing varying percentageof inhomogeneities.

We carry out some investigations on the orientation of “inhomogeneous” springs in thespecimen to study favourable and unfavourable conditions for large deformation in the mate-rial due to different orientations of inhomogeneities. We observe maximum deformation inthe case where spring elements representing nonhomogeneities are arranged in the directionof force rather than in other orientations. However actual inhomogeneities in the material arescattered in random directions. To represent random orientations, we develop random choiceof elements using uniformly distributed random numbers and assign very low stiffness valuesto those spring elements. Typically mild steel consists of about 0.9% inhomogeneities byweight. We calculate the volumetric percentage of the inhomogeneity by knowing the den-sities of pure base material (ferrous) and steel, which varies from 1.2% to 2.4%. Keeping inmind the volumetric percentage of inclusions, we vary the quantity of inclusion from 1% to4% in our model by generating uniformly distributed random numbers. Comparison betweenpercentage of inclusion and strain is tabulated in table 1. From the table we can observethat though the percentage of inhomogeneity increases, there is no great increase in the totaldeformation of the material as expected.

Figure 6 shows the effect of various percentages of inhomogeneity leading to large localstresses. In figure 7 we try to separate out springs representing inhomogeneities from thoserepresenting homogeneous material. Points very close to thex-axis show the number and thedistribution of inhomogeneities. Points due to other springs are distributed along the 45o line.It can be seen that some elements experience compressive stresses during tensile loading,which is to be expected.

Again the output of these studies does not bring out the expected behaviour of material(mild steel) under high load as we do not observe any softening or yield behaviour in the load

Table 1. Effect of extent of inhomogeneity on strain.

Extent of inhomogeneity (%) 0∗ 1 2 3 4

Strain 0.0798 0.08 0.0805 0.0815 0.8115∗Homogeneous material


−100 −−50 0 50 100 150−100

−50

0

50

100

150

200

Homogeneous material

Inho

mog

eneo

us m

ater

ial Material with 1% inhomogeneities

−100 −50 0 50 100 150−100

−50

0

50

100

150


Inho

mog

eneo

us m

ater

ial

Material with 2% inhomogeneities

−100 −50 0 50 100 150−100

−50

0

50

100

150


Inho

mog

eneo

us m

ater


−100 −50 0 50 100 150−100

50

0

50

100

150


Inho

mog

eneo

us m

ater


Compressive force

Tensile force

Compressive force

Tensile force

Compressive force

Tensile force

Compressive force

Tensile force

(d)(c)

(a) (b)

Figure 7. Distribution of spring forces for varying percentage of inhomogeneities.

verses displacement graphs. And the results are not much improved as compared to the earliercase. Thus inhomogeneities as a whole is not enough to describe material behaviour. Thenthe question is, what governs the observed material behaviour, i.e. yield under high load? Isit material instability?

3.2 Effect of material instability

The foundation of the theory of stability was laid by Euler more than two centuries ago forthe case of compressive loading. But the concept of material instability remained unnoticedfor a long time during the development of the elasto-plastic theory. Based upon Euler’s theoryand using some new concepts, Panovko & Gubanova (1965) discuss some of the materialinstabilities in tensile loading which are commonly observed.

3.2a Material nonlinearity: As we have seen, material damping and nonlinear strain-displacement relations do not cause yielding or failure but only stiffen the material at highloads. The addition of inhomogeneities in our model does not cause yielding either. The exper-imental observations show that at higher loads materials entering into the plastic zone startshowing softening behaviour and eventually get broken at some point on further applicationof load. Does this phenomenon occur due to loss of material stability under high load? Welook into material instability and obtain a relation between the applied load and the observ-able strain for a simple bar considering equilibrium in the deformed configuration, thus beingconsistent with our field equation derived earlier. Consider a bar with cross-sectional areaA,diameterd and lengthl fixed at one end and apply a uniform tensile loadF at the other endas shown in figure 8.

Diameter after deformation is given by,

dε = d(1 − νε),


F

A

l

d

Figure 8. Bar under uniaxial tensile load.

where,

ε = equilibrium strain,

ν = Poisson’s ratio,

and the cross-sectional area at equilibrium is,

Aε = (π/4) d2(1 − νε)2.

This cross-sectional area is the true or the equilibrium cross-sectional area due to consider-ation of the transverse deformation. The consideration of true cross-sectional area becomesimportant during large deformation and is equivalent to the use of the Eulerian co-ordinates.The stress acting on the body is,

σε = F/Aε,

i.e.E · ε = 4F/[πd2(1 − νε)2],

⇒ ε(1 − νε)2 = 4F/πd2E, (19)

⇒ ε0 = ε(1 − νε)2, (20)

where,E is Young’s modulus andε0 is 4F/(πEd2) which is the equivalent linear straincorresponding toF .

The behaviour ofε0 versesε is plotted in figure 9. The maximum and minimum values ofε0 andε can be determined by differentiating (20) withε and setting the derivative to zero, i.e.

∂ε0/∂ε = (1 − νε)2 − 2νε(1 − νε)

= (1 − νε)(1 − 3νε)

Therefore

∂ε0/∂ε = 0, at ε1 = 1/3ν and ε2 = 1/ν. (21)

The corresponding maximum and minimum values ofε0 can be found by substituting backvalues of ε1, 2 in (20). Thus, the maximum value ofε0 is 4/(27ν) and the minimum is 0.Let us denote the maximum value ofε0 by ε∗

0 and the corresponding value ofε by ε∗.


0 1.0 2.0 3.0 4.0 5.00

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Equilibrium strain

Equ

ival

ent l

inea

r st

rain

Figure 9. Material behaviour due to tensioninstability.

The above equation (20) is dimensionless and the material constantν is the only variable,and hence we can make a general statement that for any given homogeneous material, theequivalent linear strainε0 can not exceed the value of 4/(27ν). This is the maximum loadthat a body in tension can withstand. Beyond this point, materials are unstable. Eventually,material starts unloading until it breaks down. During this period, the deformation is likelyto confine itself to only the neck or a weak region and the remainder of the specimen wouldrecover elastically under decreasing load until fracture appears.

A similar graph can be obtained for the applied tension loadF versus the elongation atequilibrium,l, by substituting the values for Young’s modulusE and the specimen diameterd in (19). It gives a maximum load valueF ∗ for the corresponding value ofl∗ (in terms oflength scale) as shown in figure 10. We can call these two quantities the critical force andthe critical elongation respectively. It is possible to have the rod in an equilibrium state onlyup to the value ofF ∗ of force. Beyond this point, the equilibrium state is impossible for anyhigher values of the forceF . In other words, for all practical purposes, atF = F ∗ the lengthof the rod begins to increase with decrease in value of applied force which is shown by thedownward curve in figure 10, i.e. fromε = 1/(3ν) to ε = 1/ν. This range corresponds tothe non-equilibrium state of material. At the point whenε = 1/ν the material breaks i.e., thearea of the rod goes to zero which is quite obvious from figures 9 and 10. This behaviour is

0 5 10 150

0.5

1

1.5

2

2.5

3 × 105

Deformation

App

lied

incr

emen

tal f

orce

F*

* l Figure 10. Material behaviour due to tensioninstability (force vs deformation).


similar to the observation made by Panovko & Gubanova (1965) in the case of loss of stabilityof a rod in tension. However any small irregularity in the diameter will make the rod breakcloser to 4/(27ν) and also at a lower value ofε0. For example, consider a uniform rod with aneck or a irregularity created in the bar artificially by removing some of the material at someposition. Initially, the load will be carried by the entire bar. But later as the load increases,the deformation confines to the neck or irregularity zone due to the high stress concentrationobserved in these areas. In this case, the material fails at a lower load than the maximumpossible.

Thus the material instability in tension has all the characteristics necessary to describenonlinear elastic and plastic deformation and also material breakage. However, the modelpredicts this behaviour at very high strains of about 1/ν (i.e. approximately 330% in the caseof steel withν = 0.3) and therefore is not adequate to explain yielding of a steel test specimenwhich occurs around a strain of 0.1%. So we take a re-look in the next section at the behaviourof materials with inhomogeneities in the light of the instabilities discussed in this section.

Before discussing the inhomogeneities, we first extend the instability criteria discussed for1-D to 3-D. On lines similar to 1-D, we also develop a relation between equilibrium stressesversus equilibrium strains in 3-D to investigate material instability. This relation is quiteuseful to discuss instability surfaces which are similar to the classical yield surfaces. The 3-Dversion is discussed in detail in appendix A.

From the discussion on the 1-D bar, we can see that the material enters the instability zoneonce it reaches the critical equilibrium strain. At this point, the curve changes its direction(slope becomes negative) and starts descending and reaches zero at some particular value ofequilibrium strain depending upon the value ofν i.e. at 1/(3ν). Hence we can define a newyield criterion or instability criterion as the point in the load-deformation curve, where theslope of the equivalent strain versus equilibrium strain curve becomes zero. To extend theabove to 3-D, we interpret the 1-D case as the one where dσ and dε are not correlated, i.e.different values of dε exist for the same value of dσ at this point. In 3-D we can write thisrelation as

dσ = Cdε, (22)

where dσ = (dσ1, dσ2, dσ3)T is a vector of incremental principal stresses anddε similarly

is the vector of incremental principal strains(dε1, dε2, dε3)T . HereC is a tensor which gives

one to one relation between incremental stresses and incremental strains in the equilibrium.The instability criterion now becomes Det[C] = 0, whereC is given by

[C] = ∂σx/∂εx ∂σx/∂εy ∂σx/∂εz

∂σy/∂εx ∂σy/∂εy ∂σy/∂εz

∂σz/∂εx ∂σz/∂εy ∂σz/∂εz

. (23)

There are no unique solutions for dε for a given dσ . The expression for each element in theabove matrixC is given in appendix A. Now we consider three different cases.

(a) εx = εx andεy = εz = −νεx : This gives the 1-D analysis discussed so far. From figure 11awe see that for zero values of strain, the determinant is one. This point corresponds to theorigin in figure 10. When the material reaches the instability point, the determinant goesto zero. Thus, this is the point where the material changes its behaviour. In the next range,we see that value of the determinant is negative till the material breaks.


(b) εx = εx, εy = εy andεz = −ν(εx + εy): This is the 2-D case in principal strains. In thisthe determinant is a surface in equilibrium strainsεx andεy . The contour on theεx − εy

plane where the determinant is zero is the instability curve corresponding to the yieldcurve in 2-D and is shown in figure 11b.

(c) εx = εx, εy = εy, εz = εz: The instability surface obtained in 3-D case is explained stepby step in figures 11c to f. Here we observe that when the values of the strains are smallas compared to 1/(3ν), the instability surface (determinant = 0 surface) does not appear.As the strain values approach 1/(3ν) material is seen entering the instability zone and weobserve the development of the instability surface for which the determinant value is zero(figure 11c). Further figures show the development of the second surface as the value ofthe determinant goes to zero at the breakage point. The region outside of the instabilitysurface is the instability zone.

3.2b Behaviour of various terms near an inclusion:To continue our search for a suitablemodel, we now discuss the stress fields around inhomogeneities. To begin with we considerinclusion/inhomogeneities as spherical in 3-D, and around this region see how the variousterms appearing in the new field equation behave. With spherical voids, Qui & Weng (1992)observed excellent agreement with exact solution under hydrostatic loading and the finiteelement calculations are in accord with tensile behaviour of material.

In the case of 3-D, the complete stress at any point around spherical inclusion is obtainedas discussed by Timoshenko & Goodier (1970) in a polar coordinate system.

These equations are later transferred back to cartesian coordinates using the transformationmatrix and the graphs are plotted. Figures 12 to 14 show the behaviour of various terms nearan inclusion. It is interesting to note that there is a sharp variation within the length of tworadii of the inclusion.

In figure 12 we see that the invariants of linear and nonlinear strains which appear in thenew field equations gradually change away from the cavity and later get stabilised at somefixed positive value. The same phenomenon can be observed with the first invariant of thestrain rate but the second invariant of the strain rate undergoes a sharp change before it isstabilised at some negative value. On the other hand, gradients of the invariants for strain andstrain rate change sharply before stabilising at zero.

Figure 13 shows a similar trend with the strains and their gradients. Figure 14 shows thebehaviour of strain rates and their gradients. Here also we see that axial and shear strain rateterms show sharp variations within a length of twice the radius before stabilising at someconstant values. As before, gradients of strain rate quantities show sharp variations beforegoing to zero.

In the case of the pressure termp we observe that the magnitude of pressure remainsconstant through the body, which means that the gradient of the pressure term is zero. A fewearlier authors have observed thatp does not contribute to plastic deformation.

We observe that nonlinear behaviour or region of instability of material occurs around thevalue of strain equal to 1/ν. But in practice, such a high value of strain is not observed duringtension testing. Typically, in the case of steel, yield behaviour occurs at around a strain of0.001 and the breaking phenomenon occurs at around 0.05 depending upon strain rate. Now,will we be able to derive this behaviour from what we learnt so far from our bar in tension?

To bring down the value of the critical load,F ∗, and the corresponding critical strain duringmaterial instabilities to within the observed range, we introduce two new material variablesm andn, which correspond to force correction factor and strain correction factor respectively.Amongst the two, strain correction factor or actually strain enhancement factor is based onthe earlier discussion on computed strains around a spherical cavity. We find there that up


(e) (f)

(c) (d)

(a) (b)

Figure 11. Development of instability surface:(a) 1-D case;(b) 2-D case;(c) 3-D case with deter-minant zero;(d) 3-D case with negative determinant;(e)3-D case with determinant approaching zero;(f) 3-D case breaking point.


1 2 3 4 5−2

0

2

4

6

8

10

12

K1, Invariant of linear strain

K2, Invariant of nonlinear strain

Distance to diameter ratio

Gra

die

nts

1 2 3 4 5−8

−6

−4

−2

0

2

4

6

J1, First invariant of strain rate

J2, Second invariant of strain rate


Gra

die

nts

−1

0

1

2

3

4

5

6

7

K1, Invariant of linear strain

K2, Invariant of nonlinear strainS

tra

in

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

J1, First invariant of strain rate

J2, Second invariant of strain rate

Str

ain

ra

te

(a) (b)

(c) (d)

Figure 12. Behaviour of invariants near inclusion: Invariants of linear and nonlinear strain(a) andstrain rate(b); Gradients of invariants of linear and nonlinear strain(c) and strain rate(d).

to 2r from the cavity the average strains are larger than the nominal applied strain. This wouldmean that for 1% volume of nonhomogeneities in the material, about 27% of the volumehas above average strain. Since the instability would see a local strain, the material behavesas if it experiences an enhanced average strain. The level of enhancement would depend onthe geometry of the inhomogeneity. A value of 3 for a spherical cavity could becomea/b

for an ellipsoidal cavity wherea is the major axis andb is the minor axis. Thus for a flatcavity wherea/b would tend to∞ , average stress can become significantly larger than thenominal stress. This effect is represented byn, while the overall load-elongation behaviourin the stable range is determined by the strains corresponding to the nominal stress and strainenhancern. The maximum load that can be applied depends on the section at which averagestresses would peak. This section acts like the notch or groove introduced in the middle ofthe tension test specimen sometimes. This factor is brought in asm.

Figure 15 shows the behaviour of a rod which is also considered as a spring, for differentvalues ofn. In the graph, ‘ep1’ represents the behaviour of an ordinary spring while ‘ep2’ isthat of the spring with different values ofn. We can see that the factorn governs the maximumstrain at break point. An increase in value ofn reduces the strain-carrying capacity of thematerial. Figure 16 shows the behaviour asm changes for a given fixed value ofn. From thegraph we see that higher values ofm lead to lower load-carrying capacity for a fixed value of


−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Exx1, Linear strain

Exx2, Nonlinear strainS

tra

in

−2

−1.5

−1

−0.5

0

0.5

1

1.5Exz1, Linear shear strain

Exz2, Nonlinear shear strain

Str

ain

1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Exx1, Linear strain

Exx2, Nonlinear strain


Gra

die

nts

1 2 3 4 5−3

−2

−1

0

1

2

3

4

Exz1, Linear shear strain

Exz2, Nonlinear shear


Gra

die

nts

(a) (b)

(c) (d)

Figure 13. Behaviour of linear and nonlinear strains near inclusion: Linear and nonlinear strain(a)and shear strain(b); Gradients of linear and nonlinear strain(c) and shear strain(d).

n. A particular type of spring element analogous to the experimentally observed maximumload and the maximum displacement range for a given material can be generated by varyingthe values ofn andm. Figure 17 shows the combined effect of all these variables.

We try to match the experimentally observed phenomenon by arranging two or more springsboth with and without correction factorsm andn in some particular order. We study threedifferent spring arrangements. In the first case, two springs are arranged in series, while theyare in parallel with each other in the second case. The third case is generated by placing aspring in parallel with a combination of two springs in series. We discuss each case in detailin the following section.

3.3 Springs in series

Consider two springsk1 andk2 each of lengthl1 andl2 arranged in series with each other. LetF be the applied force causing deformations of magnitudes11 and12 respectively as shownin figure 18.

The total deformation of the combined spring is calculated by the addition of11 and12.Let n andm be the two correction factors as discussed earlier, applied to one of the springs.


−1.5

−1

−0.5

0

0.5

1

1.5

a

h/2

g/2S

trai

n ra

te

−5

−4

−3

−2

−1

0

1

2

3

A11

A12

A13

Str

ain

rate

1 2 3 4 5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

a

h/2

g/2


Gra

dien

ts o

f str

ain

rate

1 2 3 4 5−8

−6

−4

−2

0

2

4

6

8

A11

A12

A13


Gra

dien

ts o

f str

ain

rate

(a) (b)

(c) (d)

Figure 14. Behaviour of linear and nonlinear strain rates near inclusion: Linear(a) and nonlinear(b)strain rates; Gradients of linear strain rate(c) and nonlinear strain rate(d).

The force sustained by the ordinary spring is given by,

F = F0

(1 − ν

11

l1

)211

l1, (24)

and for the other spring,

F = F0

m

(1 − nν

12

l2

)2

n12

l2, (25)

where,F0 is the force acting on each spring. We know that,

11 = l1ε1,

12 = l2ε2,

1 = l1ε1 + l2ε2.


0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ep1

Incr

emen

tal l

oad

(dim

ensi

onle

ss)

ep2 at n = 2

ep2 at n = 4

ep2 at n = 8

ep2 at n = 16

ep2 at n = 32

Equilibrium strain

Figure 15. Effect on equilibrium strain due to change in values ofn.

In the case of springs in series, to determine the effective strain due to the application of load,we introduce a new variableα such that,

11 = αlε1

12 = (1 − α)lε2

Therefore

1 = αlε1 + (1 − α)lε2

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ep2 at m = n/16

Incr

emen

tal l

oad

(dim

ensi

onle

ss)

ep2 at m = n/8

ep2 at m = n/4

ep2 at m = n/2

Equilibrium strain

Figure 16. Effect on force due to change in values ofm.


0 0.5 1 1.5 2 2.5 3 3.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ep1

Equilibrium strain

Incr

emen

tal l

oad

(dim

ensi

onle

ss)

ep2 at m = 16ep2 at m = 8

ep2 at m = 4

ep2 at m = 2

ep2 at m = 1

n = 32

Figure 17. Combined effect on spring due to change in values ofn andm.

and

ε = 1/l = αε1 + (1 − α)ε2.

The value ofα is determined by the load-sustaining capacity of each spring. In the case ofmaterials, we representα as the effect of inhomogeneities present in the specimen.

When a set of two springs are arranged in series, the load-carrying capacity of the set isgoverned by the weaker spring. Figure 19 explains this behaviour. The weaker spring quicklyreaches its maximum force value and then starts unloading. To be in equilibrium, the otherspring also unloads such that both springs carry the same amount of force. After some timethe weaker spring breaks and the other spring also gets fully unloaded. This causes someamount of residual strain in the second spring. The magnitude of residual strain depends uponthe value ofα. If the valueα is high, then the amount of residual strain is very low and inother cases it is vice versa as shown in figure 20. In the hypothetical case when the value ofα becomes 1, we get only one line starting from the origin as in figure 20 i.e. there are noresidual strains experienced by the material. In other words, the entire material is treated as

F

∆ = ∆1 + ∆2

l1

l2

k1

k2

∆2

∆1

Figure 18. Two springs in series.


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

ep1

Equilibrium strain

Incr

emen

tal l

oad

(dim

ensi

onle

ss)

ep2 at m = 8

Figure 19. Effect of a weaker spring in series.

defect-free and its load-carrying capacity becomes very high as observed by Griffith (1921) inhis experiments with fine glass fibres. (The subject is discussed in great detail by Timoshenko1986).

3.4 Springs in parallel

Consider two springs with stiffnessesk1 andk2 arranged in parallel with each other. Let1

be the deformation caused due to the application of axial loadF as shown in figure 21. The

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

n = 32, m = 8, alpha = 0.98

Equilibrium strain

Incr

emen

tal f

orce

× 104

Figure 20. Effect of two different kinds of springs in series.


F

k1k2

∆Figure 21. Two springs in parallel.

force carried by each spring is given by,

F1 = F0(1 − νε)2ε, (26)

F2 = (F0/m)(1 − nνε)2nε, (27)

whereF1 andF2 are forces experienced by each spring andF0 is the total force acting on it,n andm are the two new material variables which govern strain and load-carrying capacityof the springs respectively andε is the equilibrium strain acting on the spring. The total forcesustained by the springs in parallel is given by

F = F1 + F2.

In the case of two parallel springs, the total forceF is the sum of the force-carrying capacityof each spring. The spring with higher values ofn andm is considered the weaker spring inthe combination. So even if the weaker spring breaks, the load continues to be carried by thestronger spring. This effect is shown in figure 22 where the effect of different values ofm fora particular value ofn is presented. We can observe a lift in the total force-carrying capacity,i.e., the curve does not touch the zero force line even after the weaker spring is broken. Thisis owing to the stronger spring which is still able to carry the applied load. In general, thisbehaviour imitates the upper and lower yield point phenomena in the material. The upperyield point appears when the weaker spring enters the instability zone and it continues up tothe lower yield point where it breaks. On increasing the load, further deformation takes placein the second spring.

3.5 Combination of series and parallel springs

Let k1 andk2 be two springs arranged in series with each other. The combination of these twosprings is placed in parallel with a third springk3. Let 11 be the displacement due to springk1 and1 be the total displacement of the combined springs due to the application of loadF as shown in figure 23. In this combination, one of the springs in the series is an ordinaryspring and other two are modified springs.

Figure 24 describes step-by-step the observed behaviour of the combined springs. In fig-ure 24a we see the individual behaviour of ordinary springs and the modified spring in seriesseparately. Figure 24b gives the combined effect of these two springs for a given value ofα.Figure 24c shows the individual behaviour of the spring which is in parallel, while figure 24dgives the combined effect of all three springs which is similar to the commonly observed


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

m = n

Incr

emen

tal l

oad

(dim

ensi

onle

ss)

m = n/2

m = n/4

m = n/8

m = n/16

ordinary spring

n = 64

Equilibrium strain

Figure 22. Combined effect of two different kinds of springs in parallel.

experimental curve for strain versus applied load. The final shape of the curve and the strainrange within which it acts depend upon the values ofm, n andα.

3.6 Summary

3.6a Field equations: Development of a new field equation and its dimensionless form in§2 and the subsequent material behaviour observed in the earlier part of this section show thatthere are three different kinds of displacement fields acting on a body. The first causes scalinguniform increase or decrease of density in the material. The final shape of the material remainsthe same but only the size varies. This effect results in a uniform stress field throughoutthe material. Hydrostatic tension or compression causes such deformation and we observevolumetric strain.

The second displacement field is one which changes the shape of the material after defor-mation without changing its density. Such deformation occurs due to the deviatoric part ofthe stress field. These stress fields do not cause any volumetric strain in the material and thetotal volume remains constant.

The third displacement field causes only rotation inside the material and does not resultin any physical deformation. These displacements come from the stresses occurring on theinterior and exterior boundaries of the material.

The various displacement fields mentioned above could be present simultaneously in amaterial. These fields are constrained by the continuity equation. More work is required toeffectively use these fields to obtain solutions to the field equations.

3.6b Material inhomogeneity and instability: In this section we see that material behaviourunder load can be represented by some typical arrangement of springs and dampers. Addi-tion of damping element helps to stabilise the mathematical model under high load. Thenonlinear nature of the springs observed during the case of instability tries to soften thematerial. Microscopic inhomogeneities cause nonuniform stress distribution in the material


F

k1

k2

∆2

∆1

k3

Figure 23. Two series springs in parallel with a third spring.

due to development of stress concentration in the vicinity of discontinuity or inhomogene-ity. It is also observed that the orientation of inhomogeneity governs the final deformation.Normally, mild steel consists of around 0.9 wt % or 1.2 to 2.4 vol % of inhomogeneity.The cause and effect of instability is brought out in this section. It appears that large straindeformation in the materials is due to the instabilities occurring inside the material. Therepresentation of instability in the material using springs can be brought to the experimen-tally observed range by varying the values of two spring correction factorsm andn andthe different types of spring arrangements discussed. The model is also able to explain the

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

Spring in series

ordinary spring

Equilibrium strain

Incre

me

nta

l fo

rce

0 0.02 0.04 0.060

0.01

0.02

0.03

0.04

0.05

0.06

0.07Combined effect of two springs in series

Equilibrium strain

Incre

me

nta

l fo

rce

0 0.02 0.04 0.060

0.005

0.01

0.015

0.02

0.025

parallel spring

Equilibrium strain

Incre

me

nta

l fo

rce

0 0.02 0.04 0.060

0.01

0.02

0.03

0.04

0.05

0.06

0.07

effective spring

n1 = 32, m1 = 8, n2 = 64, m2 = 24, alpha = 0.5

Equilibrium strain

Incre

me

nta

l fo

rce

(a) (b)

(c) (d)

Figure 24. Step by step evolution of material behaviour under tensile load.(a) Individual behaviourof the ordinary and the modified spring in series;(b) Combined effect of ordinary and the modifiedsprings;(c) Individual behaviour of the spring in parallel;(d) Combined effect of all three springs.


observations from Griffith’s experiments with thin wires. The practically observed materialphenomena which normally occur in sheet metal-forming processes like Bauschinger effect,material anisotropy, yield point phenomenon, stress inhomogeneity, solid solution strengthen-ing, strengthening from fine particles, fracture, residual stresses, etc. are explainable with thisnew model.

4. Reformulation

4.1 Field equation

The understanding of material behaviour in §3 shows that material deformation is only dueto changes in size and shape. These two factors are governed by strain and strain rate. Thefield equation developed in the earlier section consists of resistance due to three components:one is strain, the second is the dissipative part due to strain rate and the third is the pressureterm. In the case of solids the pressure term appears as a hydrostatic stress acting on thebody. From classical mechanics, it is known that deformation occurrs because of continuousaction of hydrostatic and deviatoric stresses. So going back to Lamb (1945) for the originalformulation and considering only small strain, we get

p′1 = −p + λ1(ε

′1 + ε′

2 + ε′3) + λ2ε

′1,

p′2 = −p + λ1(ε

′1 + ε′

2 + ε′3) + λ2ε

′2,

p′3 = −p + λ1(ε

′1 + ε′

2 + ε′3) + λ2ε

′3, (28)

whereε′1, ε′

2, ε′3 are the strains along each principal axes. Now, let us redefine the constant

λ2 as,

p′1 + p′

2 + p′3 = λ2(ε

′1 + ε′

2 + ε′3).

Therefore from the definition ofp we get,

p = (p′1 + p′

2 + p′3)/3 = λ2

(ε′

1 + ε′2 + ε′

3/3)

(29)

Substituting forp in (28) and rearranging the terms we get,

P′e = λ1(ε

′1 + ε′

2 + ε′3)I + λ2

ε′

1 0 00 ε′

2 00 0 ε′

3

− ε′

1 + ε′2 + ε′

3

3I

, (30)

which is in the form of total stress on the body expressed by the hydrostatic and deviatoricstresses. The values ofλ1 andλ2 can be calculated from uniaxial tension and torsional tests.From the torsional test we get,

p′1 = τ, p′

2 = −τ, p′3 = 0,

ε′1 = τ/2G, ε′

2 = −τ/2G, and ε′3 = 0,

whereτ is shear stress. Substituting in (30) and solving forλ1 andλ2 we get,

λ2 = 2G (31)


whereG is modulus of rigidity. From the uniaxial tension test,

p′1 = σ, p′

2 = p′3 = 0,

ε′1 = σ/E, ε′

2 = ε′3 = −νσ/E.

Substituting in (30) and solving forλ1 we get,

λ1 = K, (32)

whereK is bulk modulus. Thus (30) becomes the resistance due to small strains as,

P′e = K(ε′

1 + ε′2 + ε′

3)I + 2G

ε′

1 0 00 ε′

2 00 0 ε′

3

− (ε′

1 + ε′2 + ε′

3)

3I

. (33)

Similarly, the resistance due to rate of deformation can be determined. Considerε′1, ε′

2, ε′3

as the rate of strain in each direction in the principal coordinate system. Now, we can rewrite(30) for strain rate terms by substituting the strain termsε′

1, ε′2, ε′

3 by ε′1, ε′

2, ε′3 with two

more new constantsλ3 andλ4. The value of new constantλ4 can be found using the torsiontest on a thin tube specimen.

pd1 = −τ, pd2 = τ, pd3 = 0,

ε′1 = −τ/2µ, ε′

2 = τ/2µ, ε′3 = 0,

⇒ λ4 = 2µ, (34)

whereµ is material viscosity term in the case of solids. The remaining unknown constantλ3

is considered to be one more viscosity term or secondary viscosity termµ′ (Lamb 1945) inthe case of solids. Thus the resistance due to strain rate is,

P′d = µ′ d

dt(ε′

1 + ε′2 + ε′

3)I + 2µd

dt

ε′

1 0 00 ε′

2 00 0 ε′

3

− (ε′

1 + ε′2 + ε′

3)

3I

.

(35)

To evaluate the constantsλ1, λ2, λ3, λ4, we consider only small strain conditions. At smallstrains the values of elastic constant are not affected by the material nonlinearity and instabilitywhich occur at higher loads. The values ofK andG can be obtained experimentally and arestandard for a given material. Similarly the value ofµ can be calculated from the torsionalvibration of a tube. The value ofµ′ can be calculated from the longitudinal vibration of a thinwire of the material.

Considering that deformation is continuous under the action of strain and strain rate, thetotal resistance for the deformation can be determined by adding the resistance from strainand strain rate terms. Thus adding (33) and (35) we get,

P′ = P′e + P′

d

=(

K + µ′ d

dt

)(ε′

1 + ε′2 + ε′

3)I

+ 2

(G + µ

d

dt

) ε′

1 0 00 ε′

2 00 0 ε′

3

− (ε′

1 + ε′2 + ε′

3)

3I

. (36)


This expression holds good for infinitesimal strain conditions where volume change is neg-ligible. But during large deformation, the volume does not remain constant (Bell 1990). Insuch cases the change in volume should be taken into consideration. Therefore the hydro-static pressure should be expressed in terms of volumetric strain or, in other words, in termsof strain invariants. Thus for large deformation, the above equation is modified as,

P′ =(

K + µ′ d

dt

)(S ′

1 + S ′2 + S ′

3)I

+ 2

(G + µ

d

dt

) e1 0 0

0 e2 00 0 e3

− S ′

1 + S ′2 + S ′

3

3I

, (37)

wheree1, e2, e3 are the principal stretches as defined by Fung (1965) andS ′1, S ′

2 and S ′3 are

the three invariants of the principal stretches,

S ′1 = e1 + e2 + e3,

S ′2 = e1e2 + e2e3 + e3e1,

S ′3 = e1e2e3.

To observe the material behaviour under external force, consider the elastic resistance part.The above equation reduces to,

P ′ = (3K − 2G)

(S ′

1 + S ′2 + S ′

3

3

)I + 2G

e1 0 0

0 e2 00 0 e3

. (38)

From the uniaxial tension test we get, σ 0 0

0 0 00 0 0

= (3K − 2G)

(S ′

1 + S ′2 + S ′

3

3

)I + 2G

e1 0 0

0 e2 00 0 e3

, (39)

whereσ is the principal stress. In the case of a bar which is symmetric along thez-axis,e2 = e3. Thus the change in volume is given by,

S ′1 + S ′

2 + S ′3 = e1 + 2e2 + e2(2e1 + e2) + e1e

22. (40)

By solving (39) we get,

σ = 2G(e1 − e2). (41)

In the case of material instability as explained by Panovko & Gubanova (1965), we get therelationship between the instability function(1 + e2)

2 andσ/G as,

p = A(1 + e2)2σ

= 2A(1 + e2)2G(e1 − e2),

⇒ p/2AG = (1 + e2)2(e1 − e2). (42)

Now, we can plote1 ande2 as a function ofσ/G. The value ofe1 can be calculated by assumingsome negative values fore2 and solving (41) using (40) forσ/G. These plots are as shown


in figures 25 to 28. In figure 25a we can see thate2 increases linearly withσ/G at very lowload as it is only a function ofS ′

1. With increase in applied force, the curve shows nonlinearbehaviour as the remaining two invariants of strain i.e.S ′

2 andS ′3 increase their contributions.

At very high applied force, it becomes flatten (figure 26a). That is, on the further increase inload, the increase in strain in lateral direction is very small. Thus we see reduction in crosssectional area but this does not go to zero as compared to our earlier observations. At thesame time we observe thate1 increases linearly with applied load (figure 25b) and remainslinear at very high load also (figure 26b). From figure 25c and 26c we see that the strain ratioe1/e2 varies linearly with increase in load.

At the same time, the change in volume proportional to load confined to earlier deformation(figure 25d), later on further increase of load, the change in volume keeps gotting smaller(figure 26d) but does not go to zero as in earlier observations.

The instability function, behaves nonlinearly with axial strain (figure 27) but the amountof nonlinearity is small compared to earlier observations (figure 28).

Similar behaviour can be observed in the case of compressive loading. Figures 29 to 32shows the plots for cases of compressive loading.

It is important to note that in both the tensile and compressive cases the graphs are notexact mirror images of each other at the 135◦ axis line. The numerical values for each caseare different for the given range of loading.

These graphs shows the continuous relation between stress and stretch and do not showsany sign of breaking. We also observe that the change in volume remains almost constantduring large deformation.

We hypothesise that the breaking of the material therefore occurs at some critical load whena surface crack appears. The theoretical value for this breaking critical load for a material isapproximately given byE/π (Anderson 1995) based on surface energy consideration. But inpractice this value is far lowe ouring to the presence of inhomogeneities.

0 0.5 1 1.5 2 2.5−0.2

−0.15

−0.1

−0.05

0

Sigma/G

e2

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Sigma/G

e1

0 0.5 1 1.5 2 2.5−5

−4.5

−4

−3.5

−3

Sigma/G

e1/e

2

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Sigma/G

I1 +

I2 +

I3

TENSILE LOADING (a) (b)

(c) (d)

Figure 25. Behaviour of strain at small tensile load.(a) Second, and(b) first principal stretch;(c)Ratio of first and second principal stretch;(d) volumetric strain.


0 100 200 300 400 500−1

−0.8

−0.6

−0.4

−0.2

0

Sigma/G

e2

0 100 200 300 400 5000

50

100

150

200

250

Sigma/G

e1

0 100 200 300 400 500−250

−200

−150

−100

−50

0

Sigma/G

e1/e

2

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

Sigma/G

I1 +

I2 +

I3


(c) (d)

Figure 26. Behaviour of strain at higher tensile load.(a) Second, and(b) first principal stretch;(c) Ratio of first and second principal stretch;(d) volumetric strain.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

e1

Inst

abili

ty fu

nctio

n

−0.2 −0.15 −0.1 −0.05 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

e2

Inst

abili

ty fu

nctio

n


Figure 27. Behaviour of instability at small tensile load.(a) First and(b) second principal stretch vsinstability function.


0 50 100 150 200 2500

0.5

1

1.5

2

2.5

e1

Inst

abili

ty fu

nctio

n

−1 −0.8 −0.6 −0.4 −0.2 00

0.5

1

1.5

2

2.5

e2

Inst

abili

ty fu

nctio

n


Figure 28. Behaviour of instability at higher tensile load.(a) First and(b) second principal stretchvs instability function.

0 0.5 1 1.50

0.05

0.1

0.15

0.2

Sigma/G

e2

0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

Sigma/G

e1

0 0.5 1 1.5−3.4

−3.2

−3

−2.8

−2.6

−2.4

Sigma/G

e1/e

2

0 0.5 1 1.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Sigma/G

I1 +

I2 +

I3

COMPRESSIVE LOAD (a) (b)

(c) (d)

Figure 29. Behaviour of strain at small compressive load.(a) Second, and(b) first principal stretch;(c) Ratio of first and second principal stretch;(d) volumetric strain.


0 2 4 6 80

0.5

1

1.5

2

Sigma/G

e2

0 2 4 6 8−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Sigma/G

e1

0 2 4 6 8−3

−2.5

−2

−1.5

−1

−0.5

Sigma/G

e1/e

2

0 2 4 6 8−3

−2.5

−2

−1.5

−1

−0.5

0

Sigma/G

I1 +

I2 +

I3

COMPRESSIVE LOADING (a) (b)

(c) (d)

Figure 30. Behaviour of strain at higher compressive load.(a) Second and(b) first principal stretch;(c) Ratio of first and second principal stretch;(d) volumetric strain.

−0.5 −0.4 −0.3 −0.2 −0.1 0−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

e1

Inst

abili

ty fu

nctio

n

0 0.05 0.1 0.15 0.21

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

e2

Inst

abili

ty fu

nctio

n


Figure 31. Behaviour of instability at small compressive load.(a) First and(b) second principalstretch vs instability function.


−1.5 −1 −0.5 0−30

−25

−20

−15

−10

−5

0

e1

Inst

abili

ty fu

nctio

n

0 0.5 1 1.5 2−30

−25

−20

−15

−10

−5

0

e2

Inst

abili

ty fu

nctio

n


Figure 32. Behaviour of instability at higher compressive load.(a) First and(b) Second principalstretch vs instability function.

In the triaxial state of stress case, the stress acting on surface is given by total force appliedon a cross-sectional area. Therefore the stresses acting on each surfaceP′

e of a infinitesimaltetrahedron is given by,

P′e =

σ1/(1 + e2)(1 + e3) 0 0

0 σ2/(1 + e3)(1 + e1) 00 0 σ3/(1 + e1)(1 + e2)

= 1

(1 + e1)(1 + e2)(1 + e3)

σ1(1 + e1) 0 0

0 σ2(1 + e2) 00 0 σ3(1 + e3)

. (43)

Let I = S ′1 + S ′

2 + S ′3 be the change in volume. Therefore the above equation can be

written as,

P′e = 1

1 + I

σ1(1 + e1) 0 0

0 σ2(1 + e2) 00 0 σ3(1 + e3)

. (44)


Substituting (44) in (38) and writing each terms separately we get,

σ1 = (1 + e2)(1 + e3)

[(3K − 2G)

I3

+ 2Ge1

],

σ2 = (1 + e3)(1 + e1)

[(3K − 2G)

I3

+ 2Ge2

],

σ3 = (1 + e1)(1 + e2)

[(3K − 2G)

I3

+ 2Ge3

]. (45)

From the above equations we can write the first, second and third invariants of the stresstensor in terms of the strain tensor and the volumetric strain as,

21 = (3K − 2G)I + (2(3K − 2G)I/3 + 2G) S ′1

+ ((3K − 2G)I/3 + 2G) S ′2 + 6GS ′

3, (46)

22 = (1 + I)[3(2G − 3K)2 (I/3)2

+ (((2G − 3K)I/3 − 4G) (2G − 3K)I/3) S ′1

+ 4G (G − (2G − 3K)I/3) S ′2 + 12G2S ′

3

](47)

23 = (1 + I)2[(3K − 2G)3 (I/3)3 + 2G(3K − 2G)2 (I/3)2 S ′

1

+ 4G2(3K − 2G) (I/3) S ′2 + 8G3S ′

3

](48)

Thus by knowing strain and volumetric strain we can calculate the stress and stress invariants.Similarly we calculate strain and strain invariants by knowing stress and volumetric strainfrom (44) as,

a′ = (1 + I)(3K − 2G)I/3 − σ1

σ1 − 2G(1 + I),

b′ = (1 + I)(3K − 2G)I/3 − σ2

σ2 − 2G(1 + I),

c′ = (1 + I)(3K − 2G)I/3 − σ3

σ3 − 2G(1 + I)(49)

Now, we can write strain invariants in terms stress and volumetric strain as,

I ′1 = (1 + I)(3K − 2G)I/3 − σ1

σ1 − 2G(1 + I)+ (1 + I)(3K − 2G)I/3 − σ2

σ2 − 2G(1 + I)

+ (1 + I)(3K − 2G)I/3 − σ3

σ3 − 2G(1 + I), (50)


I ′2 = (1 + I)(3K − 2G)I/3 − σ1

σ1 − 2G(1 + I)· (1 + I)(3K − 2G)I/3 − σ2

σ2 − 2G(1 + I)

+ (1 + I)(3K − 2G)I/3 − σ2

σ2 − 2G(1 + I)· (1 + I)(3K − 2G)I/3 − σ3

σ3 − 2G(1 + I)

+ (1 + I)(3K − 2G)I/3 − σ3

σ3 − 2G(1 + I)· (1 + I)(3K − 2G)I/3 − σ1

σ1 − 2G(1 + I), (51)

I ′3 = (1 + I)(3K − 2G)I/3 − σ1

σ1 − 2G(1 + I)· (1 + I)(3K − 2G)I/3 − σ2

σ2 − 2G(1 + I)·

(1 + I)(3K − 2G)I/3 − σ3

σ3 − 2G(1 + I). (52)

4.2 Final form of the equation

During large deformation, the small strain tensors are not able to provide the complete geo-metrical interpretation. Fung (1965) used the concept of stretch to interpret large deformationas,

e1 = 1 − (1 − 2ε1)1/2,

where he consideredε1 as a small strain component. However, here we treat it as a componentof the Almansi strain tensor. After some simplification we see that both stretch and strain areof the same order.

Equation (37) gives the final form of the resistance to deformation in the principal coordinatesystem. Using the transformation matrix, and going back to the spatial coordinate system,

P = LT P ′L

=(

K + µ′ d

dt

)(S1 + S2 + S3)I + 2

(G + µ

d

dt

)(E − S1 + S2 + S3

3I

),

(53)

whereE is the stretch tensor in the spatial coordinate system andS1, S2, S3 are the corre-sponding invariants.

E = I − (I − 2E∗)1/2,

S1 = Eii ,

S2 = − 12(EijEij − EiiEjj ),

S3 = det(E).

Now substitutingP in (10), the law of balance of linear momentum in Eulerian coordinateand simplifying, we get the final form of the equation of motion as,

ρDUDt

= ρB + ∇T

[(3K − 2G + (3µ′ − 2µ)

d

dt

)(S1 + S2 + S3

3

)]

+[∇ ·

(2

(G + µ

d

dt

)E)]T

. (54)


We observe that the above equation has only three variables, i.e. three displacements alongthree directions. The density term can be expressed in the form of volumetric strain aswe have seen during the earlier formulation. The pressure termp does not appear in thisformulation. Thus there is no one more equation, the continuity equation, as we did ear-lier. The change in material density during the deformation automatically takes care of themass conservation contrary to that assumed in the case of fluid. Fluid is expected to deformlarge by under shear load and therefore the effect of volume change is separately related tohydrostatic pressure during deformation. Thus an additional equation is required to take intoaccount the conservation of mass as a supplementary to the Navier–Stokes equation for flu-ids. This however is avoided in our final formulation above. As discussed earlier, the fourunknown material constantsK, G, µ′ andµ can be obtained from simple measurements.The equations are applicable from solids to fluids and therefore could form the basis for dis-cussing plasticity of solids by including their microscopic inhomogeneities as well as fordiscussing turbulence in fluids by including their microscopic structures as found in flowingfluids.

5. Conclusions

The finalised field equation is obtained in the Eulerian coordinate system using Almansistrain and Cauchy stress tensors without any major assumptions. The developed relationshows that deformation occurs due to continuous contribution from strain and strain rateterms. Under the action of load, material behaves like a compressible fluid and change involume occurs due to changes in strain invariants. The addition of the elastic effect in theformulation considers residual stresses and spring-back actions in the case of elasto-plasticdeformation. Elastic unloading naturally occurs in our formulation. Resistance to non-elasticdeformation comes from gradients of rate of deformation. The approach strengthens thetheory put forth by microscopic modelling of plastic deformation that local gradients causedby defects and microscopic inhomogeneities determine the plastic behaviour of materials.Instabilities during tension tests are brought out by the model. The role of inclusions andmicroscopic inhomogeneities in plastic flow can also be observed. The model explains theclassical understanding of the high strength of materials drawn into thin wires. Bauschingereffect, material anisotropy, yield point phenomenon, effect of stress inhomogeneities, andfracture are easily explained by the model. It also helps to understand material strengthening,ductility and failure phenomena.

Appendix A

Consider an equilibrium system with the forcesσx , σy andσz acting along thex, y andz direction respectively in the principal coordinate system. Letεx , εy and εz be the cor-responding strains as shown in figure A1. Now consider the situation where a bodyundergoes deformation in all three directions. We can write stresses acting at a point inequilibrium as,

σx = Eεx(1 + εy − ν(εx + εz))(1 + εz − ν(εx + εy)),

σy = Eεy(1 + εz − ν(εy + εx))(1 + εx − ν(εy + εz)),

σz = Eεz(1 + εx − ν(εz + εy))(1 + εy − ν(εz + εx)). (A1)


dy

y

z

dz dx

x

∈

∈

∈

Figure A1. Strains in a body under equilibrium.

Let ε1 , ε2 andε3 be the first, second and third invariants of strain. i.e.,

ε1 = εx + εy + εz,

ε2 = εxεy + εyεz + εzεx,

ε3 = εxεyεz. (A2)

Adding equations forσx, σy andσz and rearranging the terms, we get the first invariant ofstress2′

1 as,

2′1 = E[ε1 − 2νε2

1 + ν2ε31 − 2(1 + ν)ε2 + 2ν(1 − ν)ε1ε2 + 3(1 + ν)2ε3],

(A3)

where

2′1 = σx + σy + σz.

The second invariant of stress22 can be obtained by,

2′2 = E2(σxσy + σyσz + σzσx),

substituting from (A1) in the above equation, expanding and rearranging the terms, we get,

2′2 = E2[ε2 − (1 + ν)ε1ε2 − 3ν(1 − ν)ε2

1ε2 + ν2(1 − ν)ε31ε2 − ν3ε4

1ε2

+ 3(1 + ν)4ε23 + (1 + ν)2ε2

2 + 3(1 + ν)(1 − 2ν)ε1ε3 − 2ν(1 + ν)2ε1ε22

− 3ν(2 − ν)(1 + ν)ε31ε3 + ν2(1 + ν)2ε2

1ε22 + 3ν2(1 + ν)ε3

1ε3

+ 2(1 + ν)3ε2ε3 − 4ν(1 + ν)3ε1ε2ε3 + 3(1 + ν)ε3]. (A4)


Similarly, the third invariant of stress2′3 can be determined,

2′3 = σxσyσz

= E3ε3[1 + (1 − 2ν)ε1 + ν(ν − 2)(ε21 − 2ε2) − ν(1 − ν + ν2)(ε1ε2 − 3ε3)

+ (1 − 2ν + 3ν2)ε3 + (1 + 3ν2 − 2ν3)ε2 + ν2(ε31 − 3ε1ε2 + 3ε3)]

2 (A5)

The elements of the matrixC which appear in (23) of §3 while developing the instabilitycriteria are given by,

σx/εx = −νεx(1 + εz − ν(εx + εy)) − νεx(1 + εy − ν(εx + εz))

+ (1 + εz − ν(εx + εy))(1 + εy − ν(εx + εz)),

σx/εy = εx(1 + εz − ν(εx + εy)) − νεx(1 + εy − ν(εx + εz)),

σx/εz = −νεx(1 + εz − ν(εx + εy)) + εx(1 + εy − ν(εx + εz)),

σy/εx = εy(1 + εz − ν(εx + εy)) − νεy(1 + εx − ν(εy + εy)),

σy/εy = −νεy(1 + εz − ν(εx + εy)) − νεy(1 + εx − ν(εy + εz)),

+ (1 + εz − ν(εx + εy))(1 + εy − ν(εx + εz)),

σy/εz = −νεy(1 + εz − ν(εx + εy)) + εy(1 + εx − ν(εy + εz)),

σz/εx = εz(1 + εy − ν(εx + εz)) − νεz(1 + εx − ν(εy + εz)),

σz/εy = −νεz(1 + εy − ν(εx + εz)) + εz(1 + εx − ν(εy + εz)),

σz/εz = −νεz(1 + εy − ν(εx + εz)) − νεz(1 + εx − ν(εy + εz))

+ (1 + εy − ν(εx + εz))(1 + εx − ν(εy + εz)).

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