Strain energy of the elements and can be written as:

(7–275)

where:

= = elastic strain vector

= total strain vector =

= thermal strain vector defined in Equation 7.18

= stress vector =

= stress-strain relationship matrix (constitutive matrix)

Figure 7.33: Stress Vector Definition

The stress is related to the strains by:

(7–276)

In Equation 7–275, the stress-strain relationship of the isotropic model can be written using strain tensor as follows:

(7–277)

where:

(7–278)

From Equation 7–277, the stress-strain relationship of the isotropic model can be rewritten in the matrix form:

(7–279)

This may also be inverted to:

(7–280)

(7–281)

where:

The table below shows representative values for densities and elastic constants for the major isotropic materials.

From Equation 7–275 to Equation 7–277, the stress-strain relationship of the orthotropic model can be rewritten in the matrix form:

(7–282)

where typical terms are:

(7–283)

The parameters v_yx, v_yz and v_xz must be defined by the user. The D^-1 matrix is presumed to be symmetric, the values of v_yx, v_zx and v_zy are calculated as:

(7–284)

(7–285)

(7–286)

Thermal materials exhibit changes in shape, area, and volume in response to temperature variations. Properties such as the coefficient of thermal expansion, thermal conductivity, specific heat, and reference temperature are crucial for calculating the nodal temperature and resulting thermal strain from heat transfer.

Thermal materials are classified based on their behavior under temperature changes into two main categories: linear thermal materials and non-linear thermal materials.

 

7.2.2.3.1. Linear Thermal Materials

Linear thermal materials are characterized by their constant thermal properties over a significant range of temperatures.

The primary properties include:

Coefficient of Thermal Expansion (α): This measures how much a material expands per unit change in temperature, typically remaining constant for linear materials. The coefficient is applied in all directions of an element. The formula is given by:

(7–287)

where:

L = original length

= rate of change in length per unit change in temperature

Thermal Strain (ε^th): As the material undergoes a temperature change, it experiences a thermal strain, calculated by:

(7–288)

where:

ΔT = change in temperature from the reference temperature

Thermal Conductivity (k): It quantifies the material's ability to conduct heat, expressed as:

(7–289)

where:

q = heat flux

k = thermal conductivity

∇T = temperature gradient

Specific Heat (c): This property indicates the amount of heat per unit mass required to raise the unit temperature and is defined as:

(7–290)

where:

C = heat capacity

ρ = density

V = volume of a body

The reference temperature is typically set based on the material’s application, providing a baseline for thermal expansion and other calculations.

The above thermal properties are used to construct the governing equation for Thermal Analysis.

 

7.2.2.3.2. Non-linear Thermal Materials

Non-linear thermal materials are characterized by properties that change with temperature. Unlike linear materials, where properties such as Young's modulus E, the coefficient of thermal expansion α, thermal conductivity k, and specific heat c are constant, non-linear materials require these properties to be defined as functions of temperature such as E(T), α(T), k(T), and c(T), typically provided in the form of temperature-value curves. The thermal strain ε^th is calculated by the formula:

(7–291)

where:

α(T) = coefficient of thermal expansion as a function of temperature T

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Note:   In the input data, the x-axis represents temperature, and the y-axis represents the values of each parameter dependent on temperature.

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The form Neo-Hookean strain-energy potential is:

(7–292)

where:

C10 = material constant

D = material incompressibility parameter

J = volume strain

Ii = principal invariants of the left Cauchy-Green deformation tensor

= modified invariants, calculated as:

(7–293)

The relationship between initial shear modulus μ₀ and C₁₀ is:

(7–294)

where:

The initial bulk modulus is related to the material incompressibility parameter by:

(7–295)

where:

The form of the strain-energy potential for Arruda-Boyce model is:

(7–296)

where:

μ = initial shear modulus of material

λM = limiting network stretch (locking stretch)

D = material incompressibility parameter

J = volume strain

= first invariant of Cauchy-Green deformation tensor

The Ogden form of strain-energy potential is based on the principal stretches of left-Cauchy strain tensor, which has the form:

(7–297)

where:

n = the number of terms in the summation of the Ogden model

μk, αk and Dk = material constants

= deviatoric principal stretches at the ith principal axis which can be calculated with the eigenvalue of the Cauchy-Green deformation tensor of λi as follows:

(7–298)

The initial shear modulus, μ is given as:

(7–299)

The initial bulk modulus K is defined by:

(7–300)

Remarks

The strain-energy potential of the Ogden compressible foam model defined in the MAPDL solver is:

(7–301)

where:

and = material constants (user input values in the Engineering Data of Ansys Workbench)

In the form of the above formula, the following relationship is established:

The Motion solver only supports two-term Mooney-Rivlin model. The form of the strain energy potential for a two-parameter Mooney-Rivlin model is:

(7–302)

where:

C10 and C01 = material constants

D = material incompressibility parameter (user input)

J = volume strain

= first invariant of Cauchy-Green deformation tensor

= second invariant of Cauchy-Green deformation tensor

The initial shear modulus is given by:

(7–303)

where:

The initial bulk modulus is:

(7–304)

where:

The nominal strain-stress curves for BILINEAR and MULTILINEAR models are shown in the figure below:

Figure 7.35: Nominal Strain-Stress Curve for Plasticity Materials

where:

If the nominal stress is equal to an initial yielding stress σ_y0, the material will develop plastic strains. If it is less than σ_y0, the material is elastic and the stresses will develop according to the elastic stress-strain relations and the formulation of the material is identical to that of a linear material. The strain-stress relationship in the plastic region is defined by tangent modulus E_t. For tangent modulus, as this value becomes zero, the material is close to the perfect plasticity. In this case, the Motion solver will more iterate to find the yielding stress.

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Note:  When using the Motion Standalone Preprocessor for multilinear plasticity materials, the strain-stress relationship in the plastic region is defined by a spline. The x-axis of the spline represents the differences in strain, Δε, and the y-axis represents the nominal stress, σ_i as shown in Figure 7.5: Definition of Multilinear Strain-Stress Curve for Motion Standalone Preprocessor. The differences in strain, Δε_i, are defined as:

(7–305)

where:

εi = ith nominal strain in the nominal strain-stress curve

n = total amount of data for the multilinear plasticity material

When using Workbench environment, Strain-Stress curves defined in the Engineering Data are automatically converted to Δε_i and σ_i and written to the solver input file.

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The hardening rule describes the changing of the yield surface with progressive yielding, so that the conditions (stress states) for subsequent yielding can be established. Two hardening rules are available: work (or isotropic) hardening and kinematic hardening. In work hardening, the yield surface remains centered about its initial center line and expands in size as the plastic strains develop. For materials with isotropic plastic behavior this is termed isotropic hardening and is shown in Figure 7.37: Types of Hardening Rules (a). Kinematic hardening assumes that the yield surface remains constant in size and the surface translates in stress space with progressive yielding, as shown in Figure 7.37: Types of Hardening Rules (b).

Figure 7.37: Types of Hardening Rules

Motion support two types of linear kinematic hardening: Beta (β) and Prager linear hardening.

The total strain is defined as:

(7–306)

where:

The option for Plastic Strain in the analysis setting determines whether to calculate the total strain as true strain or as engineering strain.

The plastic hardening modulus is calculated from the following equation.

(7–307)

where:

The yielding stress and plastic strain can be calculated as the unknown variables in the Motion solver.

The Cauchy stress of the plasticity material can be expressed using stretch as follows:

(7–308)

where:

The nominal stress can be calculated with the strain as follows:

(7–309)

where f(ε_i) can be determined from the strain-stress curve.

 

7.2.4.2.1. Beta (β) Model

Beta (β) is a model that linearly represents the fundamental concepts of isotropic hardening, kinematic hardening, and combined hardening, which is an intermediate state between the two. The hardening rule according to the value of β is as follows:

(7–310)

Figure 7.38: Hardening Rule for the Beta (β) Model

The corresponding yield stress can be defined as follows, increasing with plastic deformation:

(7–311)

For a more detailed explanation of isotropic hardening and kinematic hardening, refer to Isotropic Hardening and Kinematic Hardening in the Material Reference.

 

7.2.4.2.2. Prager Linear Hardening

The Prager linear hardening model is a specific case of the Chaboche kinematic hardening model. The Chaboche model is an advanced plasticity model that accounts for both isotropic and kinematic hardening through a series of nonlinear equations. The model's general form can be expressed as follows:

(7–312)

where:

x = back stress

ci, γi, i = 1,2,…,n = material constraints for kinematic hardening

The back stress x is superposition of several kinematic models as:

(7–313)

where:

n = number of kinematic models to be superposed

In Prager linear hardening, γ is assumed to be 0, and the number of kinematic models is 1. Thus:

(7–314)

or equivalently,

(7–315)

For a more detailed explanation of Chaboche kinematic hardening, refer to Nonlinear Kinematic Hardening in the Material Reference.

Remarks

To obtain the same solution as when β=1 using the Prager linear hardening model, use c and E_t as follows:

(7–316)

(7–317)

where:

= value corresponding to Et when Beta is used for the linear kinematic hardening type

To obtain the same solution as when β=0 using the Prager linear hardening model, use c and E_t as follows:

(7–318)

(7–319)

When using plasticity materials, the determination of whether the stress has exceeded the yield stress is evaluated at the element's integration points. If the stress calculated at the Gauss points is extrapolated to the nodal points, the nodal stress may be calculated to be higher than that at the Gauss points. This could cause confusion by suggesting that the material has begun to undergo plastic deformation at points where the nodal stress significantly exceeds the yield stress.

To avoid this confusion, the Motion solver provides the following three options:

Figure 7.39: Strain-Stress Report for Plasticity Materials