Constitutive Modeling(ppt)

Response of nonlinear elastic models is similar to the proverbial “… the last straw which broke the camel’s back.”
Elastoplasticity
Certain basic steps are followed in formulating all constitutive models based on plasticity. One must define: • yield criterion/surface • plastic potential/flow rule • hardening rule
dε = dε + dε
• For example in 1D:
e
p
1 1 dε1 = dε + dε = + d σ1 E H
e 1 p 1
where E=elastic Young’s modulus, and H=plastic modulus
Hardening rule
• Describes the evolution of the yield surface during loading.
Summary of elements of elastoplasticity:
Additivity: Elasticity:
dε = dε e + dε p
dσ = D : dε
e
p
e
∂g (σ, q) Flow rule: dε = ϕm (σ, q) = ϕ ∂σ
Hardening rule:
Example of a failure surface: Mohr-Coulomb:
Plastic potential/Flow rule
• Flow rule gives directions of plastic strain increment – usually taken as normal to a plastic potential.
Motivation for developing advance constitutive models for geomaterials “Advances in computer technology and computational methods provide incentives for developing complex models for the constitutive behavior of geomaterials.”
n = ∂f ∂σ
ξ = ∂f ∂q
Derivation of incremental relations:
dε = dε e + dε p dσ = D e : dε e
(a) (b) (c)
Rearrangement of (g) gives ϕ:
ϕ= n : D e : dε n : De : m − ξ : h
included in most models
anisotropy (inherent and induced) principal stress rotation creep, stress relaxation and rate effects three-phase system (air, water, solids) instability, bifurcation and strain localization discontinuity and heterogeneity
dσ = Dep : dε
(ktion in terms of ϕ:
df = n : D e : (dε − ϕm ) + ϕξ : h = 0
(l)
(g)
In (l), D ep is the fourth-order elastoplastic constitutive stiffness tensor:
f (σ, q) = 0
• A stress point cannot lie outside the yield surface.
• A yield surface must lie within or, at most, be coincident with the failure surface. Their shapes are usually taken to be similar.
∂g (σ, q) dε = ϕm(σ, q) = ϕ ∂σ
p
• If f=g then flow is associative, otherwise flow is nonassociative.
Additivity
• Total strain consists of elastic and plastic components.
[D ] = [C ]
ijkl ijkl
−1
Nonlinear elastic models
Two of the most widely used: 1)Duncan-Chang hyperbolic model 2)Nelson and Barron K-G model • Can be used in simple monotonic loading cases. • Relates strain increment to stress increment (dσ and dε are co-axial). • Cannot model deformations after local failure/plastic yield unless principal stress directions remain nearly constant.
units of [C] = 1/stress units of [D] = stress
σ ij = Dijkl ε kl ⇒ {σ } = [D ]{ε }
In 3D Cijkl and Dijkl each has each 34 = 81 elements (constitutive parameters). Note that:
Constitutive Modeling of Geomaterials
Marte Gutierrez, PhD Associate Professor Department of Civil & Environmental Engineering Virginia Polytechnic Institute & State University Blacksburg, VA 24061
(i)
Substitution of (d) and
dq = ϕh(σ, q)
dq =
(e)
n : D e : dε h n : De : m − ξ : h
(j)
to Prager’s consistency condition
df = n : dσ + ξ : dq = 0
(f)
De : m )( n : De ) ( e : dε dσ = D − e n D m ξ h − : : :
Elements of realistic constitutive models for geomaterials:
• • • • • • • • • • • nonlinear stress-strain behavior stress dependent stiffness and strength dilation/contraction due to pure shear irrecoverable deformation stress-path dependency
Yield criterion/surface
• Divides stress space into two parts - inside the yield surface only elastic (reversible) strains occur. Outside both elastic and plastic strains occur.
F (σ ) = 0
Examples of yield surfaces: Cam clay: Modified cam clay:
f = q + Mp ln ( p pc )
f = q 2 − M 2 p ( pc − p )
1 + sin φ 2c cos φ σ1 = σ3 + 1 sin − φ 1 − sin φ
f (σ , q ) ≤ 0
ϕ≥0
fϕ=0
To Karush-Kuhn-Tucker form is a stronger requirement than Prager’s consistency condition which states that for plastic loading:
df = n : dσ + ξ : dq = 0
d q = ϕ h (σ , q )
Types of hardening: 1) Non-hardening (perfect plasticity) 2) Isotropic hardening (uniform expansion of yield surface) 3) Kinematic hardening (translation of yield surface(s) – tangency at point where surfaces meet. 4) Isotropic-kinematic 5) Distortional (almost not used)
CHILE = Continuous, Homogenous, Isotropic, Linear Elastic DIANE = Anisotropic, Inhomogenous, Anisotropic, Nonlinear, Elastoplastic
ε ij = Cijklσ kl ⇒ {ε } = [C ]{σ }
d q = ϕ h (σ , q )
Karush-Kuhn-Tucker form:
To obtain the plastic multiplier ϕ giving the magnitude of the plastic strain increment, the following conditions must be satisfied:
(h)
dε p = ϕm(σ, q)
Substitution ϕ into (d):
dσ = D
e
Combine (a) to (c) to yield:
dσ = D e : dε − ϕD e : m
(d)
(D : dε −
e
: m )( n : De )
n : De : m − ξ : h
: dε
neglected in most models
Types of Stress-Strain Laws
σ
ε Linear elastic (follows Hooke’s Law)
Non-linear Elastic
σ
σ
ε Elastic Perfectly Plastic
ε Cyclic Hardening Plasticity (Non-linear Elastoplastic)
Outline of presentation
• Some background materials • Review of elastoplasticity formulation • 3D models for geomaterials • Example 1 – liquefaction of sand • Example 2– constitutive for soft rock time dependent behavior and rock fluid interaction • Areas of future research, outstanding issues