Differential equation of equilibrium:
In Moldex3D Stress analysis, the following assumptions are made:
(1) The materials of parts are linearly elastic
(2) The strains and displacement are small
(3) The behavior of structure is static and linear
The behavior of a material is said to be linearly elastic when the strain is linearly proportional to the stress, and the geometry would return to its undeformed state when the loads are removed.
Under the above conditions, the following governing equilibrium equations are used:
σij, j + fi = 0
where σij denotes the stress components and fi is the body force.
Material constitutive models:
For linearly elastic materials, stress is proportional to the strain which is indicated below,
σij = cijklεkl
where cijkl denotes the elastic constants and εij is the strain components.
The above equation is an approximation of stress that is valid as the strain quantity is located within the elastic range. The above relation is known as the constitutive equation. Different types of materials will have different forms of constitutive equations as explained in the following.
A material is said to be isotropic if its properties are identical in all directions. For isotropic materials, the constitutive equations take the form:
σij = λεkkδij + 2μεij
Where λ and μ are Lame’s coefficients, relating to Young’s modulus E and Poisson’s ratio v by
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The properties of anisotropic materials are directional dependent. For anisotropic material, the constitutive equation is usually expressed in matrix notation, also called the Voigt notation, as follows:
{σ}=[C]{ε}
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Moldex3D Stress module applies the Finite Element Method (FEM) to discretize the governing equation. In the FEM, the solution region is discretized into many small, interconnected units or finite elements. This allows a complicated model to be approximated by many finite elements. In each element, a convenient approximate solution is assumed and the conditions of overall equilibrium are derived. An approximate solution can be obtained by satisfying these conditions.
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A mesh is an assembly of elements and nodes. Elements are defined and connected by nodes. Several types of elements are used in Moldex3D Stress module, including: 4-node tetrahedral element, 5-node pyramidal element, 6-node prismatic element, and 8-node hexagonal element, as illustrated below.
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Matrix Solver
After the description of model and construction of the equilibrium equations, a set of linear algebraic equations are collected and solved using a matrix solver. In Moldex3D Stress module, the multi-grid accelerated iterative methods are adopted as the default solver.
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