Presentation of the

General Linear Strength Theory (GLST)

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Strength and Engineering Journal

of the “Collegium” All World Academy of Sciences

Munich (Germany)

10 (2010), 1

This report is dedicated to the memory of my dear teacher,

Academician Georgy Stepanovich Pisarenko,

to the 100th anniversary of his birthday

G. S. Pisarenko Institute for Problems of Strength

of the National Academy of Sciences of Ukraine

is unique in strength science development.

Scientific schools of

• Academician G. S. Pisarenko who created and

developed this Institute as its first director,

• his successor Academician V. T. Troshchenko,

• Academician A. A. Lebedev,

• and their prominent colleagues

are well-known

GLST REASONS

• Linearity provides extreme simplicity, usability, & further

investigation & development possibilities

• Generalization & applicability range determination

of known linear strength criteria, e.g.

• greatest normal stress criterion (Galilei, Rankine)

- σ_c ≤ σ₃ ≤σ₁ ≤ σ_t

• greatest normal strain criterion (Mariotto, de Saint-Venant)

σ₃ - ν(σ₁ + σ₂) ≥ - σ_c , σ₁ - ν(σ₂ + σ₃) ≤ σ_t (ν is Poisson's ratio)

• greatest shear stress criterion (Tresca)

σ_e = σ₁ - σ₃ ≤ σ_L

• the same with a uniform term (a is a constant) (Davidenkov)

σ_e = σ₁ - σ₃ + a(σ₁ + σ₂ + σ₃) ≤ σ_L

• single-shear strength theory (Coulomb, Mohr)

σ_e = σ₁ - ασ₃ ≤ σ_t (α = σ_t / σ_c)

SOME KNOWN POSSIBILITIES OF

LINEAR STRENGTH CRITERIA

Yu’s twin-shear unified strength theory (b is a constant)

σ_e = σ₁ - α(bσ₂ + σ₃)/(1 + b) ≤ σ_t

if σ₂ ≤ (σ₁ + ασ₃)/(1 + α)

σ_e = (σ₁ + bσ₂)/(1 + b) - ασ₃ ≤ σ_t

if σ₂ ≥ (σ₁ + ασ₃)/(1 + α)

Its piecewise linear strength criteria family provides

• considering σ₂ (σ₁ ≥ σ₂ ≥ σ₃), + (σ, σ, σ), relation (τ_L , σ_t , σ_c)

• fitting strength test data on many materials

• implementing in commercial simulation programs

GLST SCIENTIFIC FOUNDATIONS

• Constructive philosophy

Principles of

• tolerable simplicity,

• unity,

• universality, etc.

• Elastic mathematics

Fundamental data processing methods

correcting and generalizing

• the relative error and

• the least square method

which both have many lacks of principle

FUNDAMENTAL STRENGTH SCIENCE

• general theory of stresses reduced to individual limits

σ_j° = σ_j / |σ_jL| (j = 1, 2, 3)

which are reciprocal individual reserves with signs

• universality principles for σ_j° & strength criteria in σ_j°

• general theories of considering σ₂ , + (σ, σ, σ), relation (τ_L , σ_t ,

σ_c), anisotropy, & loading variability via vectorial σ_j°

• general theory of complex equivalent stresses σ_e , σ_e°

which can take negative and imaginary values, too,

with introducing |σ_e| , |σ_e°|

• stress sign unification in mechanics & geomechanics

via introducing pressures

p = - σ , p₁ = - σ₃ , p₂ = - σ₂ , p₃ = - σ₁ , p₁ ≥ p₂ ≥ p₃

2D diagrams of 3D (limiting) stresses & pressures

x , y , z (σ₁ = σ₂ = σ₃)

σ_m = (σ₁ + σ₂ + σ₃)/3

-p_m = - (p₁ + p₂ + p₃)/3

σ_d = p_d = (σ_x² + σ_y²)^1/2

By no rotational symmetry: any section, σ_d‘= σ_dσ_dL‘/σ_dL

• general reserve theory: n = σ_dL/σ_d

both by constant σ_m if σ_m ≤ 0

and by the constant direction to the origin if σ_m ≥ 0

GLST INITIAL FORM

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° +

Σ_i=1^N b_1i|c_00i + c_11iσ₁° + c_21iσ₂° +

c_31iσ₃° + b_2i|c_02i + c_12iσ₁° + c_22iσ₂° + c_32iσ₃° + … + b_Hi|c_0Hi +

c_1Hiσ₁° + c_2Hiσ₂° + c_3Hiσ₃°|| ... | ≤ 1 (with moduli nesting)

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° +

Σi₌₁^N b_i|c_0i + c_1iσ₁° + c_2iσ₂° + c_3iσ₃°| ≤ 1.

By strength data scatter, N = 0 & N = H = 1 suffice

• Exhaustivity methodologically important

• Convenience & excluding mistakes due to a general method of

representing piecewise functions via unified (single) formulae

by any real-number functions and nesting levels H of moduli

Very particular cases

• Tresca’s criterion: σ_e = (|σ₁ - σ₂| + |σ₂ - σ₃| + |σ₃ - σ₁|)/2 ≤ σ_L

• Yu’s twin-shear unified strength theory: σ_e = ((2 + b)σ₁ + (1 -

α)bσ₂ - α(2 + b)σ₃ + b|σ₁ - (1 + α)σ₂ + ασ₃|)/(2 + 2b) ≤ σ_t

GENERAL PURE LINEAR STRENGTH CRITERIA (N = 0)

• Initial form (with constants a_j):

σ_e° = a₀ + a₁σ₁° + a₂σ₂° + a₃σ₃° ≤ 1

• Final form (with any real constant a):

σ_e° = σ₁° + aσ₂° - σ₃° ≤ 1

• Correction of the universalization of Tresca’s

criterion due to adding the simplest particular case of

f(σ₂°) with f(0) = 0

• For any strength criterion

σ_e° = F(σ₁°, σ₂°, σ₃°) ≤ 1

possibly nonlinear, its correction is

σ_e° = F(σ₁°, σ₂°, σ₃°) + f(σ₂°) ≤ 1

• Pyramidal form of the limiting surface

adequate by a > 0

• Generalizing σ_e , σ_e° with using values

σ_e , σ_e° < 0 if necessary

• Considering σ₂ , + (σ, σ, σ), & relation (σ_t , σ_c)

• Predefined relation (τ_L , σ_t , σ_c):

1/τ_L = 1/σ_t + 1/σ_c (τ_L = σ_L/2 if σ_t = σ_c = σ_L)

like the Coulomb-Mohr (Tresca) strength criterion

• Monotonic dependence of σ_e , σ_e° on σ₂ , σ₂°

contradicting some strength criteria and

available strength test data on many materials

UNIVERSALIZATION OF TRESCA’S CRITERION (a = 0)

σ_e° = σ₁° - σ₃° ≤ 1

• σ_e = σ₁ - σ₃ ≤ σ_c if σ₁ ≤ 0

like a model material with σ_tm = σ_cm = σ_Lm = σ_c

• σ_e = σ₁ - ασ₃ ≤ σ_t (α = σ_t / σ_c) if σ₃ ≤ 0 ≤ σ₁

the Coulomb-Mohr linear criterion, applicability domain

• σ_e = σ₁ - σ₃ ≤ σ_t if σ₃ ≥ 0

like a model material with σ_tm = σ_cm = σ_Lm = σ_t

Remark. Pure one-modulus criteria give nothing

new by a = 0 and lead to limited both σ_ttt and

σ_ccc with contradicting known strength tests

GENERAL MIXED LINEAR STRENGTH CRITERIA (N = 1)

σ_e°= a₁σ₁°+ a₂σ₂°+ a₃σ₃°+ b|c₁σ₁°+ c₂σ₂°+ c₃σ₃°| ≤ 1

(initial homogeneous form)

Key role of

• sign[τ_L/σ_t - σ_c/(σ_t + σ_c)], or

• sign(τ_L/σ_L - 1/2) by σ_t = σ_c = σ_L ,

with b = τ_L(σ_t + σ_c)/(σ_tσ_c) - 1

Final form

σ_e° = (1 - (τ_L(σ_t + σ_c)/(σ_tσ_c) - 1)|c₁|)σ₁° +

aσ₂° - (1 - (τ_L(σ_t + σ_c)/(σ_tσ_c) - 1)|c₃|)σ₃° +

(τ_L(σ_t + σ_c)/(σ_tσ_c) - 1)|c₁σ₁° + c₂σ₂° + c₃σ₃°| ≤ 1

• For materials with τ_L/σ_t = σ_c/(σ_t + σ_c)], or

τ_L/σ_L = 1/2 by σ_t = σ_c = σ_L ,

there are additional strength criteria

using any c₁ and c₃ with c₁c₃ ≤ 0 and any b

Initial nonhomogeneous form

σ_e° = a₀ + a₁σ₁°+ a₂σ₂° + a₃σ₃° +

b|c₀ + c₁σ₁°+ c₂σ₂°+ c₃σ₃°| ≤ 1,

for such materials only, additional criteria

- bc₀ + (1 - bc₁)σ₁° + a₂σ₂° + (-1 - bc₃)σ₃° +

b|c₀ + c₁σ₁° + c₂σ₂°+ c₃σ₃°| ≤ 1

GENERAL POWER STRENGTH THEORY

Homogeneous shears powers

(generalizing Hosford’s criterion) (k > 0)

σ_e = [a₁₃(σ₁ - σ₃)^k + a₁₂(σ₁ - σ₂)^k + a₂₃(σ₂ - σ₃)^k]^1/k ≤ σ_L (k > 0).

Uniaxial tension and compression give

σ_e = {a(σ₁ - σ₃)^k + (1 - a)[(σ₁ - σ₂)^k + (σ₂ - σ₃)^k]}^1/k ≤ σ_L

Pure shear (σ₁° = τ_L/σ_Lt , σ₂° = 0 , σ₃° = -τ_L/σ_Lc) gives (k ≠ 1)

σ_e° = {(σ_Lt^kσ_Lc^k/τ_L^k - σ_Lt^k - σ_Lc^k)/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k](σ₁° - σ₃°)^k +

[(σ_Lt + σ_Lc)^k - σ_Lt^kσ_Lc^k/τ_L^k]/[(σ_Lt + σ_Lc)^k - σ_Lt^k - σ_Lc^k][(σ₁°- σ₂°)^k + (σ₂° - σ₃°)^k]}^1/k ≤ 1

GENERAL QUADRATIC STRENGTH THEORY

n = 2:

σ_e° = {[σ_Ltσ_Lc/(2τ_L²) - (σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc)](σ₁° - σ₃°)² +

[(σ_Lt² + σ_Lc²)/(2σ_Ltσ_Lc) - σ_Ltσ_Lc/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1

Additionally σ_Lt = σ_Lc = σ_L :

σ_e° = {[σ_L²/(2τ_L²) - 1](σ₁° - σ₃°)² + [2 - σ_L²/(2τ_L²)][(σ₁°- σ₂°)² + (σ₂° - σ₃°)²]}^1/2 ≤ 1

Critical particular cases of τ_L/σ_L

• τ_L/σ_L = 1/2, a = 1:

σ_e° = σ₁° - σ₃° ≤ 1 (Tresca’s criterion)

• τ_L/σ_L = 1/3^1/2, a = 1/2:

σ_e° = {[(σ₁° - σ₃°)² + (σ₁° - σ₂°)² + (σ₂° - σ₃°)²]/2}^1/2 ≤ 1

the Huber-von Mises-Hencky criterion

• τ_L/σ_L = 1/2^1/2, a = 0:

σ_e° = [(σ₁° - σ₂°)² + (σ₂° - σ₃°)²]^1/2 ≤ 1

GENERAL SYMMETRIC POLYNOMIAL CRITERIA

σ_e° = [∑_i=0^N a_iP_i(σ_1n°, σ_2n°, σ_3n°)]^1/N ≤ 1

where N is a positive integer,

P_i(x, y, z) are homogeneous symmetric

polynomials of power i

N = 2:

σ_e° = [σ_1n°² + σ_2n°² + σ_3n°² - a(σ_1n°σ_2n° + σ_1n°σ_3n° + σ_2n°σ_3n°) +

bf°(σ_1n° + σ_2n° + σ_3n°)]^1/2 ≤ 1

where function f = 0 by standard tests

Piecewise linear function f(σ₁ + σ₂ + σ₃) =

• σ₁ + σ₂ + σ₃ + σ_Lc if σ₁ + σ₂ + σ₃ ≤ -σ_Lc ,

• 0 if -σ_Lc ≤ σ₁ + σ₂ + σ₃ ≤ σ_Lt ,

• σ₁ + σ₂ + σ₃ - σ_Lt if σ₁ + σ₂ + σ₃ ≥ σ_Lt .

Limiting surface type

• paraboloidal by nonzero f

• ellipsoidal by zero f and -2 < a < 1

• cylindric by zero f and a = 1 (Huber-von

Mises-Hencky)

• hyperboloidal by zero f and a > 1

GENERAL CONCLUSIONS

• The general formulae unification

method using moduli simply represents

real-number functions piecewise

defined

• Stresses reduced to their individual

limits are universal and suitable for

strength data processing

• Strength criteria in the reduced stresses

can be universal mechanical laws of

nature

Relation between shear & normal strengths

Influence of σ₂ and pressure on strength

• Considering this relation and this influence

should and can be independently provided

• For considering this relation critical for

selecting strength criteria form, simple linear

(with one modulus) and quadratic functions

usually suffice

• For considering this influence, adding simple

functions (vanishing by standard tests) either

of σ₂ or, for symmetry, of σ₁ + σ₂ + σ₃ suffices

• Quadratic strength criteria with linearly

considering this influence provide paraboloidal

limiting surfaces

GENERAL LINEAR STRENGTH THEORY

• uses suitable single formulae

possibly with moduli

• has clear physical sense

• exhaustively represents and generalizes all the

piecewise linear strength criteria

• determines their applicability domains

• correctly considers material science phenomena

• provides generally representing and processing

test data

• fits strength test data on materials very different

• provides initial strength criteria for fundamental

material strength science