Tensor

Class object is a handler of invariant tensor object that is represented by components and basis at any time.

Base class container may contain of arrays with any dimension and rank (even rank = 0 for scalar arrays). Class container is inherited from std::array, so it is fast. Based on container class object provides basic tensor-vector and scalar arrays calculus in N dimensional spatial.

Class object provide +/-/scal product operators between each other. So you should not keep in mind at how basis are component of current tensor/vector, internal functionality does operation correctly.

Using smart pointers for manage basis and component containers allows to have many objects (tensor/vector) at the same basis object. So all objects linked to the one basis will change at the same time when basis will be changed. Basis object will live till the last tensor/vector object will be destroyed.

There is possibility to handle array of scalar or even just scalar variable like tensor object but without basis poiter. It is an advantage for abstraction and construction of any measures (see the next paragraph).

Measure

The main purposes the next features are imtroduced for fast implementation of any mathematical model based on state variable approach. The main points:

• state of a system may be described by finite set of tensor or/and scalar state variables (SV)
• each state variable may have evolutionary equation (usually - differential equation)
• each state variable may have dependecies on other strate variables

Class StateMeasure based on object class is providing unified logic for working with state variable:

• it has rate - value of rate of SV
• it has value - value SV
• also it contains previous value and rate of SV for ability to implement of more convinience numerical schemas
• rate_equation(T t, T dt) - evolution equation in rate form (the rate of state variable dependency on set of parameters and set of SV's)
• finit_equation(T t, T dt) - evolution equation in finite form (describes dependency of finite value on set of parameters and set of SV's)
• calc_rate(T dt) - by default it is the first order schema to calculate rate, may be overriden
• integrate_value(T dt) - by default the first order (Euler) schema to integrate value, may be overriden

Implemented measures

There are a few predefined wide used strain and stress measures:

• deformation gradient GradDeform, it is a base measure in a Finite strain therory, it also contains methods to calculate derived measures (left, right Hencky measures, stretch and Cauchy tensors and etc)
• Cauchy stress measure CaushyStress, it is a base measure Caushy stress from wich (and GradDeform) my be derivated another stress measures like Piola–Kirchhoff tensor.

Numerical schema

Base abstract class AbstractSchema describes main functionality:

• init() - assumed that it is called before calc rate/finite and integration procedure
• calc() - the main stage of numerical schema, rate or value on the end of current step are calculated
• finalize() - actions after if necessery

There are a few types of numerical schema:

• RATE_CALCULATE: dX(n+1) := F(...), X(n+1) = X(n) + dX(n+1)dt
• FINITE_CALCULATE: X(n+1) := G(...), dX(n+1) = (X(n+1)-X(n))/dt

Class StateMeasureSchema implements (abstract methods of AbstractSchema) a plain first order numerical schema over StateMeasure.

Material point

Class MaterialPoint is base for any material model. it is inherited from AbstractSchema and contains basis object (common for any measures), so any tensor measures are linked to the same basis, at the same time indifferent scalar measures are not linked to any basis.

Models

Any class inherited from class MaterialPoint is a material model contains array of StateMeasureSchema's, logic of any neccesary calculation and Relation links two or more StateMeasureSchema's.

Relation

In case of dependency on StateMeasureSchema from other (or others) derived classes are implemeted:

• ElasticRelation (inherited from StressMeasure, template from StressMeasure and StrainMeasure) implemets elastic relation between StressMeasure and StrainMeasure (for example Hooke's law). It is extended stress measure of the material model (so it has polymorphic behavior as stress measure and relation).
• PlasticRelation (inherited from StrainMeasure, template parameters - StressMeasure and StrainMeasure) implements plastic flow rule (dependecy plastic strain part on stress). It is extended plastic strain measure of the material model (so it has polymorphic behavior as plastic strain measure and plastic relation).
• StrainDecomposition (inherited from StrainMeasure, template parameter - StrainMeasure) implements full strain measure decomposition into plastic and elastic ones.
• GeometricNonlinearity implements dependency the basis orientation of MaterialPoint on the state. It may be a part of any model if necesessary.

Base Elastic Model

Elasticity inherited from MaterialPoint (template parameters - StressMeasure and StrainMeasure) and implements elastic behavior. It has extended ElasticRelation instead of pure StressMeasure (it is both stress measure and relation with dependency on strain measure).

Inelastic (plastic) Model

Class Plasticity inherited from Elasticity (template parameters - StressMeasure and StrainMeasure) and implements plastic behavior. It has PlasticRelation instead of pure plastic part of Strain<easure (it is both strain measure and relation with dependency on stress measure). Also it has StrainDecomposition as a rule for an elastic part calculation (full and plastic parts are known).