Stress Analysis in Viscoelastic Materials MEEN 5330, Fall 2006 Presented by: SURESH KUMAR ANBALAGAN SAI PRASEN SOMSETTY VEERABHADRA REDDY KAJULURI JAMAL MOHAMMED SUBRAHMANYA SRI VITTAL CHEDERE MEEN 5330

Introduction Hooke’s Law Viscoelastic Material Relation with Hooke’s Law Examples MEEN 5330

Some phenomena in viscoelastic materials A) Instantaneous elasticity B) Creep under constant stress C) Stress relaxation under constant strain D) Instantaneous recovery E) Delayed recovery F) Permanent set MEEN 5330

CREEP Slow progressive deformation of a material under constant stress Anelastic Material MEEN 5330

THREE STAGES OF CREEP MEEN 5330

STRESS RELAXATION MEEN 5330

Basic Elements These models allow the mathematical representation of viscous and elastic properties of viscoelastic materials. Basic elements: (Spring and Dashpot) F = -KX F is restoring force by spring, K is spring constant and X is spring elongation. F/A = -(K/A)*X σ = E ε E is modulus of elasticity. Spring. MEEN 5330

Basic Elements Piston cylinder arrangement with a perforated bottom A viscous lubricant between the cylinder and piston walls force ‘F’ is proportional to the velocity F = K (dX/dt) σ = η (dε / dt) η is Viscosity coefficient Dashpot MEEN 5330

Maxwell model For series connection the total strain is given by ε = ε1 + ε2 strain rate is given by ε *=ε1*+ε2* ε *= σ*/E + σ /η ε * is (dε/dt) Spring and dashpot in series MEEN 5330

Kelvin/Voight Model σ = σ1+ σ2 (total stress) σ1 = E ε (for spring) σ2 = η ε* (for dashpot) σ = E ε + η ε* Spring and Dashpot in parallel MEEN 5330

Generalized Maxwell model The generalized Maxwell model consists of a series of Maxwell model as shown below, MEEN 5330

Generalized Maxwell model cont’d The total strain is given by MEEN 5330

Generalized Maxwell model cont’d Consider the following setup of Maxwell model units, MEEN 5330

Generalized Maxwell model cont’d The above diagram shows a generalized model of Maxwell model units connected in parallel. The total strain is given by, MEEN 5330

Example for a Generalized Maxwell model consider the following arrangement of Maxwell model units. Determine the stress-strain relations for the given arrangement. MEEN 5330

Example for a Generalized Maxwell model cont’d Solution For a generalized Maxwell model, we have the stress – strain relation as follows MEEN 5330

Example for a Generalized Maxwell model cont’d For N=2, the above equation becomes, We know the relaxation time relation as, MEEN 5330

Example for a Generalized Maxwell model cont’d Using the above relation the equation for the strain changes to, Which when expanded becomes MEEN 5330

Generalized Kelvin Model The generalized Kelvin model is represented using a number of Kelvin model units connected in series as shown MEEN 5330

Generalized Kelvin Model cont’d The total strain, for an N number of Kelvin model units is given as follows MEEN 5330

Standard Linear Solid For a spring, σ = E1 ε1 ε1* = σ*/E1 For the Kelvin model, σ = E2 ε2 + η2 ε2* η2 ε2* = σ - E2 ε2 ε2* = 1/ η2 (σ – E2ε2) three parameter linear solid model MEEN 5330

Standard Linear Solid cont’d for a series combination of linear solid, ε = ε1 + ε2 strain rate ε* = ε1* + ε2* ε* = σ*/E1 + 1/ η2 (σ – E2ε2) MEEN 5330

Three parameter Viscous model For a dashpot σ = η ε* ε1* = σ/ η1 For the Kelvin model, σ = E2 ε2 + η2 ε2* η2 ε2* = σ - E2 ε2 ε2* = 1/ η2 (σ – E2ε2) three parameter viscous model MEEN 5330

Three parameter Viscous model cont’d for a series combination of the viscous model, ε = ε1 + ε2 strain rate ε* = ε1* + ε2* ε* = σ/ η 1 + 1/ η2 (σ – E2ε2) ε* = σ (1/ η 1 + 1/ η2) - E2ε2/ η2 MEEN 5330

CONCLUSION The study of the stress-strain analysis of viscoelastic materials helps us understand the procedure and the steps involved in determining the stress-strain relations for a whole bunch of different models. This in time will prove to be useful in solving any kind of stress-strain analysis problems that exhibit the kind of behavior as discussed in this report. MEEN 5330

Homework problem Determine the stress - strain equation for a four parameter model (four parameter model is a Maxwell and a Kelvin/Voight Model connected in series). MEEN 5330

REFERENCES MEEN 5330