# Example: Linear Elasticity

The weak formulation of the Lame equation describing linear elasticity is to find u in a suitable space such that

for all vector-valued test functions v from a suitable space. The assembly within ViennaFEM for is triggered by following exactly the mathematical formulation by first defining vector-valued unknowns and test functions, and then specifying the weak formulation:

Defining the weak form of the Lame equation

// Define trial and test functions std::vector< FunctionSymbol > u(3); u[0] = FunctionSymbol(0, unknown_tag<>()); u[1] = FunctionSymbol(1, unknown_tag<>()); u[2] = FunctionSymbol(2, unknown_tag<>()); std::vector< FunctionSymbol > v(3); v[0] = FunctionSymbol(0, test_tag<>()); v[1] = FunctionSymbol(1, test_tag<>()); v[2] = FunctionSymbol(2, test_tag<>()); // Define weak form: std::vector< Expression > strain = strain_tensor(u); std::vector< Expression > stress = stress_tensor(v); Equation weak_form_lame = make_equation( integral(symbolic_interval(), tensor_reduce( strain, stress )), //= integral(symbolic_interval(), viennamath::constant(1.0) * v[2]) );

Here, strain_tensor() and stress_tensor() are simple implementations of the strain and stress tensor using ViennaMath, respectively, and tensor_reduce() sums over the entries of both tensors. The remaining code is similar to the Poisson equation example.